)i\ 



AUTOGRAPHIC STRAIN- 
ILLUSTRATING THE EFFECTS OF TIME 



DIAGRAMS OF METALS, 

IND OF TEMPERATURE UPON RUPTURE, 



TESTING MACHINE OF PROFESSOR R. H. THURSTON. 




A TREATISE 



ON THE 



RESISTANCE OF MATERIALS, 

3lntt an 3tppenfcte 



ON THE 



PRESERVATION OF TIMBER. 



BY 



DE VOLSON WOOD, 



PBOFESSOK OF MATHEMATICS AND MECHANICS IN STEVENS INSTITUTE OF TECHNOLOQYt 



FOURTH EDITION, REVISE, 



NEW YORK: 
JOHN WILEY & SONS, 
15 ASTOR PLACE. 

1883. 






•n, 



* 



Entered according to Act of Congress, in the year 1875, bj 

JOHN WILEY & SON, 
In the Office of the Librarian of Congress at Washington. 






Trow's 

Printing and Bookbinding Co^ 

printers and bookbinders, 

TOi-213 East \7.tk St., 

NEW YORK. 



PREFACE. 



The subject which forms the title of this work is inexhaustible. 
Volumes have been published containing the results of experiments, 
and yet experimental investigations, especially in regard to iron, 
were probably never more numerous than at the present time. The 
infinitely varied character of the materials, and of the great variety 
of conditions under which they are used, renders it impossible for a 
limited number of experiments to cover the whole ground. The 
most refined analysis has been brought to bear upon the subject, and 
yet many problems, which at first appear to be comparatively simple, 
remain unsolved. For instance, no theory of the rupture of a simple 
beam has yet been proposed which fully satisfies the critical experi- 
menter. 

Numerous theories have been proposed from time to time in regard 
to the resistance of materials under strain, but none are universally 
satisfactory. I do not agree with Barlow's theory of rupture involv- 
ing his " Resistance to Flexure," and hence I have put all references 
to it in fine print, except the statement of its principles. 

The general plan and scope of this edition are essentially the same 
as the former one. I have, however, omitted some matter which 
appeared to be unimportant, and have added considerable new matter 
which, I trust, will add to the scientific value of the work. 

I have given considerable prominence to the subject of shearing 
stresses and strains. Shearing strains are somewhat analogous to the 
flowing of the particles over each other. As our knowledge of the 
subject becomes more critical, this branch of it becomes more im- 
portant. 



IV PREFACE. 

I have given a new formula for the deflection of a beam (Equation 
(219a) ), but I have not sufficient data at hand of the proper kind 
to test its accuracy in practice. It will doubtless be tested, sooner 
or later, when its possible accuracy will be determined. 

I have sought to present the subject in such light as to impress 
upon the mind of the student that he is learning only the rudiments, 
whilst a large field remains for time to explore. 

It is with pleasure that I here acknowledge my indebtedness to 
Professor W. A. Norton, of New Haven, Conn., and to my colleague, 
Professor It. H. Thurston, of the Stevens Institute of Technology, 
for valuable and original matter. 

DE VOLSON WOOD. 

Hoboken, February 27th, 1875. 



PREFACE TO THE THIRD EDITION. 



Since the first publication of this work, the science of the resist- 
ance of materials has made considerable progress, both experi- 
mentally and analytically. During the preparation of the second 
edition, certain experiments at the Stevens Institute of Technology and 
at other places threw so much light upon two or three points, that 
some of the statements in the latter part of the work appeared to 
conflict slightly with those in the first part. In the first edition, on 
page 41, it was implied that to break iron increased the ultimate 
strength of the separate pieces. But this result seemed so question- 
able, that I modified it in the next edition, as will be seen on page 
44 of this volume. However, before the second edition was com- 
pleted, a report of Commodore Beardslee's remarkable experiment 
was received, and which is here given on page 250. This experiment 
shows that the first statement may have been perfectly correct. 
Whether it is of practical value or not is of little consequence to 
the writer, for the most important thing in regard to experimental 
science is the determination of facts. Their practical application will 
follow sooner or later. 

The elevation of the elastic limit due to overstrain and subse- 
quent rest is one of those interesting facts in connection with iron 
which has been clearly brought out by Professor Thurston's experi- 
ments. A brief summary of some of these experiments is given in 
the latter part of this work. 

The effect of heat and cold upon the ultimate strength of iron 
and steel as deduced from more extended experiments, does not seem 
to agree with the inference drawn from the earlier experiments, but 



VI PREFACE TO THE THIRD EDITION. 

it is not safe, with our present knowledge, to draw very general con- 
clusions in regard to the effects of temperature upon them. It seems 
quite clear, however, that the quality of the metal is an important 
element in the problem. Theoretically considered, homogeneous 
Solids ought to increase in tenacity as the degree of cold increases ; 
for heat acts like a repulsive force tending to separate the particles 
from each other. 

Experiments now being made show that the reported values of 
the co-efficient of elasticity may, in extreme cases, have been errone- 
ous. It is found that ordinary iron is liable to bend when subjected 
to a severe pull. Whether this be due to a want of homogeneity of 
the material, or to imperfections in the machine, makes little or no 
difference with the result ; for, if the measurements of the elonga- 
tion be taken on the concave side, it will be too small, but if on the 
opposite side, it will be too large, and the shorter the piece the 
greater will be the per cent, of the error. The mean of several 
readings on different sides would give a reliable result. But the fact 
that this error has been detected, shows the importance of having a 
complete record of all the circumstances of the experiment, for 
scientific purposes, such as the length, form, and transverse dimen- 
sions of the part tested ; the part, if any, beyond that measured ; 
and the manner of making the measurements. I consider that any 
conclusion drawn from analysis founded upon the supposition that 
wrought iron of a quality which would be used by a good mechanic 
may have as low a coefficient of elasticity as 17,000,000 lbs., or as 
high as 40,000,000 lbs., is open, at least, to suspicion. 

A large field yet remains for experiments upon large, full-sized 
specimens, such as are used in actual construction. It is desirable 
that our government should carry forward the work which they have 
recently inaugurated in this direction. 

The progress that has been made in analytical investigations con- 
sists chiefly in the solution of special problems. The three grand 
beacon-lights in this science are Young's Modulus of Elasticity ; 
Navier's Approximate Solution of the Curve of Flexure, and Lame's 
Theorie de V Elasticity des Solides. Without the first, this subject 
ceases to be a science ; without the second, the analysis for the simplest 
problems of flexure would be so complex as to be nearly worthless for 
practical use ; and without the last, numerous theoretical problems 
would have remained unsolved. Of the problems solved since the 



PREFACE TO THE THIRD EDITION. Vll 

first publication of this work, we notice particularly the general 
problem of the continuous beam, the analytical solution of which, by 
Professor Weyrauch, of Germany, and the more practical, but scarcely 
less general solution, by Professor Mansfield Merriman, of New 
Haven, Conn., are very complete, and in both cases are extended so 
as to include framed continuous girders. 

The analysis of the " Skeleton Structure " in the form of bridges 
and roofs in common use, is sufficiently complete and comprehensive to 
satisfy the demands of both the theorist and the practical man. In 
the future the chief study will doubtless be upon the details of the 
structure. The pin, the eye, the rivet, the joint, demand more skill 
and watchful care in their mechanical execution than the larger mem- 
bers of a structure, and the ability of a piece to perform the duty im- 
posed upon it, depends as much upon the mechanical execution of its 
connection with other pieces, as upon its internal structure. The 
determination of the proportions of these small pieces in accordance 
with the principles of elasticity, often involves a high order of anal- 
ysis, and the investigations are more or less lengthy. The determi- 
nation of the proper diameter of an iron pin, in the so-called pin-con- 
nection, by Mr. Bender, C.E., and of the proper form of the eye on an 
eye-bar, by Professor Burr, both of which were published by the 
American Society of Civil Engineers, are good illustrations of what 
may be accomplished in this direction, but the analysis is too heavy 
for an elementary work upon this subject. Practical men, however, are 
not ready to accept results which follow only from such intricate anal- 
ysis, and the one making the analysis, being obliged to make cer- 
tain assumptions, generally feels better satisfied with his results if 
they agree with those deduced from carefully made experiments. If, 
however, the theories in regard to stress give results which agree 
with those determined experimentally in the great variety of forms 
which are used in practice, then will confidence be secured in the 
results of theory when applied to forms which have not been practi- 
cally tested. This is the true object of theory — to enable one to 
determine results under new conditions. 

De V. W. 
Hoboken, September, 1877. 



TABLE OF CONTENTS. 



INTRODUCTION. 

NO. OF THE 

Article. PAQH 

1. Statement of General Problems 1 

2. Definitions of Terms , 2 

3. G-eneral Effect of Stresses 3 

4. General Principles of Elastic Resistances 3 

5. Definition of the Coefficient of Elasticity 4 

6. Proofs of Certain Laws 5 



CHAPTER I. 

TENSION. 

7. Elastic Resistance to Tension ? 

8. Results of Experiments upon Wrought-Iron 8 

9. Results of Experiments upon Cast Iron 10 

10. Graphical Representation of Results 12 

11. Hodgkinson's Experiments 12 

12. Barlow's Experiments upon Malleable Iron 14 

18. Experiments of Chevandier and Wertheim upon Wood 15 

14. Elasticity of Wood Radially and Tangentially 16 

15. Remarks upon the value of E 17 

16. Problem. Elongation of a Prismatic Bar 17 

17. Problems. Elongation of a Bar and Cone 19 

18. Work of Elongation 20 

19. Vertical Oscillations. .- 21 

20. Viscosity of Solids 22 

21. Modulus of Strength 24 

22. Formula for Modulus of Strength 24 

23. Upper Section of a Suspended Bar 25 

24. Bar of Uniform Strength 26 

25. Strains in a Closed Cylinder 27 

26. Strength of Glass Globes to Internal Stress 30 

27. Riveted Plates 31 

28. Strength of Drilled and Punched Plates 3S 

29. Size and Distribution of Rivets 34 

30. Strength of Iron in different directions of the Rolled Sheet 35 

81. Tensile Strength of Boiler Iron , Stevens' Experiments 38 



X CONTENTS. 

NO. OP THE 

Article. page 

32. Tensile Strength of Wrought Iron at various Temperatures 38 

33. Strength of other Metals at different Temperatures. 42 

34. Effect of Severe Strains 42 

35. Effect of Repeated Rupture ; 44 

36. Effect of Annealing on Steel and Iron 45 

37. The Strength of Iron and Steel depends upon other conditions 45 

38. Chrome Steel 47 

39. Effect of Prolonged Fusion of Cast Iron 47 

40. Effect of Remelting 47 

41. Effect of Cooling Castings 48 

42. Modulus of Strength Modified 49 

43. Safe Limit of Loading 49 



CHAPTER II. 

COMPRESSION. 

44. Rodman's Experiments 50 

45. Compression of Wrought-Iron 52 

46. Graphical Representation 52 

47. Comparative Resistance of Cast and Wrought Iron 52 

48. Compression of Steel 54 

49. Compression of other Materials 54 

50. Example 54 

51. Example 55 

52. Parallelism of Sections not exact 56 

53. General Problem of the Equilibrium of an Elastic Solid 56 

54. Analytical Expression of Equilibrium 56 

55. Lateral Expansion accompanies Longitudinal Contraction 56 

56. Modulus of Crushing. Modulus of Strains 57 

57. Resistance to Crushing of Cast Iron 58 

58. Resistance of Wrought Iron to Crushing 59 

59. Resistance of Steel to Crushing 59 

60. Resistance of Wood to Crushing 60 

61 . Resistance of Glass to Crushing 60 

62. Strength of Pillars 61 

63. Weight of Pillars 64 

64. Condition of the Casting of Pillars 66 

65. Buckling of Tubes 66 

66. Collapse of Tubes 67 

67. Discussion of Results 70 

68. Thickness of Tubes— Law of 70 

69. Formula for Thickness of Tubes 73 

70. Elliptical Tubes 73 

71. Very long Tubes 74 

72. Comparison of Strength from Internal and External Pressure 74 

73. Collapsing of Glass Globes 74 



CONTENTS. Xl 



CHAPTER IIT 

THEOKIES OP FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 

NO. OF THE 

Akticle. PAGE 

74. General Remarks 76 

75. Galileo's Theory 76 

76. Hooke's Theory 77 

77. Mariotte's and Leibnitz's Theory 77 

78. Bernoulli's Theory 78 

79. Parent's Theory 78 

80. Coulomb's Modifications 78 

81. Young's Modulus of Elasticity 78 

82. Navier's Modifications 79 

88. The Common Theory. 79 

84. Barlow's Theory 81 

85. Transverse Elasticity 84 

86. Remarks upon the Theories 86 

87. Position of the Neutral Axis found Experimentally 86 

88. Position found Analytically 88 



CHAPTER IV. 

SHEARING STRESS. 

89. General Statement 93 

90. Modulus of Shearing Stress 93 

91. Problem of a Tie-Beam. 95 

92. Probl m of Riveted Plates 96 

93. Transverse Shearing Stress 96 

94. Bending due to Transverse Shearing Stress 99 

95. Shearing Resistance to Torsion 101 



CHAPTER V. 

FLEXURE. 

96. Equation of the Elastic Curve 104 

97. Moment of Inertia 106 

98. General Problem 107 

99. Beam Fixed at one End and Loaded at the Free End 107 

100. Beam Fixed at one End and Uniformly Loaded 109 

101. Preceding Cases Combined 110 

102. Supported Beam Loaded at any Point. Ill 

103. Supported Beam Loaded Uniformly 113 

104. The two preceding cases Combined 114 



Xll CONTENTS. 

NO. OF THB 

ARTICLE, page 

105. Beam Fixed at one End, and Supported at the other, and Loaded at 

any Point 116 

106. lb., Uniformly Loaded 122 

107. Beam Fixed at its End and Loaded at the Middle 124 

108. lb. , Loaded Uniformly 125 

109. Results Collected : 128 

110. Remarks upon the Results 128 

111. Problem. Beam Supported at Three Points 130 

112. Beam Supported at many Points 132 

113. Beams of Variable Sections 136 

114. Beams subjected to Oblique Strains 137 

115. Flexure of Columns 139 

GRAPHICAL METHODS. 

116. Graphical Method Explained 142 

117. General Problem of Deflection 142 

118. Case 1 145 

119 . Case II 146 

120. Case III 147 

121. Case IV 147 

122. Other cases 147 

123. Moment of Inertia of a Rectangle. . . 148 

124. Moment of Inertia of a Triangle 149 

125. Moment of Inertia of a Circle 149 

126. Moment of Inertia of other Surfaces 151 

127. Vibration of Beams 151 



CHAPTER VI. 

TRANSVERSE STRENGTH. 

128. Strength of Rectangular Beams . . 153 

129. Modulus of Rupture 156 

130. Practical Formulas 157 

131 . Relative Strengths of Beams 157 

132. Examples 158 

133. Relation between Strains and Deflection 159 

134. Hollow Rectangular Beams 160 

135. Double J Beams 160 

136. True meaning of d, 162 

137. Experiments of Baron von Weber 164 

138. Graphical Method, Strength of Square Beam 167 

139. Square Beam, — Diagonal vertical , 167 

140. Irregular Sections 169 

141. Strength according to Barlow's Theory 171 

142. Beams Loaded at Several Points 172 

143. Partial Uniform Load 173 



CONTENTS. xiu 

NO. OF THE 

Article. pagb 

144. General Formula 175 

145. Rectangular Beam 176 

146. Rectangular Beam with. Sides Inclined 176 

147. Strongest Rectangular Beam which, can be cut from a Cylinder 178 

148. Triangular Beams 178 

149. Trapezoidal Beam 180 

150. Cylindrical Beams 182 

151. Elliptical Beams 183 

152. Parabolic Beams 183 

153. Formula according to Barlow's Theory 184 

154. Oblique Strains 184 

155. Position of the Neutral Axis for Minimum Strength 186 

CHAPTER VIL 

BEAMS OP UNIFORM RESISTANCE. 

156. General Expression 188 

157. Beams Fixed at one End and Loaded at the Free End 188 

158. lb. , and Uniformly Loaded x 190 

159. Previous Cases Combined 192 

160. Weight of the Beam the only Load... 192 

161. Beams Supported at their Ends 194 

162. Beams Fixed at their Ends , 196 

163. Modifications due to Transverse Shearing 196 

164. Unsolved Problems # 197 

165. Forms of Cast Iron Beams 198 

166. Hodgkinson's Formulas 201 

167. Results of Experiments upon "J" Rails 202 

168. Wrought Iron Beams 202 

169. Peculiarly constructed Floor 202 

CHAPTER VIII. 

TORSION. 

170. Torsive Strains 204 

171. The Angle of Torsion 204 

172. Value of the Coefficient 207 

173. Torsion Pendulum 207 

174. Rupture by Torsion 208 

175. Practical Formulas 210 

175a. Results of Wertheim's Experiments. 211 

CHAPTER IX. 

DISTORTIONS. 

176. General Statement 214 

177. Measure of Slipping 214 



XIV CONTENTS. 

NO. OF THE 

Article. Pagh 

178. Relation between Longitudinal and Lateral Stresses 215 

179. Shearing Strains due to Transverse Stresses 223 

ISO. Deflection due to Transverse Shearing 229 

181. Deflection cue to Longitudinal Slipping 232 

] 82. Lines of Maximum and Minimum Strains 285 



CHAPTER X. 

EFFECT OF LONG -CONTINUED STRAINS — OF OFT-REPEATED STRAINS — RE- 
MARKS UPON THE CRYSTALLIZATION OF IRON. 

183. Effect of Long- continued Strains 241 

184. Hodgkinson's Experiments 242 

185. Vicat's Experiments 242 

186. Fairbairn's Experiments 242 

187. Roebling's Observations 245 

188. Fairbairn's Experiments upon Oft-repeated Strains 246 

189. Stiffening under Strains . 249 

190. Effect of Shocks 250 

191. Crystallization of Iron 256 

192. Practical Solutions 260 

193. Thurston's Experiments 263 

CHAPTER XL 

LIMITS OF SAFE LOADING OF« MECHANICAL STRUCTURES. 

194. Risk and Safety 271 

195. Absolute Modulus of Safety 271 

196. Factor of Safety 272 

197. Rational Limit of Safety 274 

198. Practical Examples 276 

1 99. Proof Load .280 

APPENDIX I. 

Timber and its Preservation. 281 

APPENDIX II. 

Solutions and Explanations 289 

APPENDIX III. 

Table of Properties of Materials. 309 



A TREATISE 



ON 



THE RESISTANCE OE MATERIALS. 



INTRODUCTION. 

l c IN PROPORTIONING ANY MECHANICAL STRUCTtBE, 

there are at least two general problems to be considered: — 

1st. The nature and magnitude of the forces which are to be 
applied to the structure, such as moving loads, dead weights, 
force of the wind, etc. ; and, 

2d. The proper distribution and magnitude of the parts 
which are to compose the structures, so, as to successfully resist 
the applied forces. . 

The former of these problems may be solved without any 
reference to the latter, as the structure may be considered as 
composed of rigid right lines. The latter depends principally 
upon the mechanical properties of the materials which compose 
the structure, such as their strength, stiffness, and elasticity, 
under various circumstances. 

The mechanical properties of the principal materials — wood, 
stone, and iron* — have been determined with great care and 
* expense by different experimenters, both in this and foreign 
countries, to which reference will hereafter be made. 

* The properties of mortars have been thoroughly discussed by Gen. Q. A. 
Gtilmore in his work on Limes, Mortars, and Cements. 1862. 



2 THE RESISTANCE OF MATERIALS. 

2. DEFINITIONS OF TERMS. 

Stresses are the forces which are applied to bodies to bring 
into action their elastic and cohesive properties. These forces 
cause alterations of the forms of the bodies upon which they 
act. 

Strain is a name given to the kind of alterations produced 
by the stresses. The distinction between stress and strain is not 
always observed ; one being used for the other. One of the 
definitions given by lexicographers for stress, is strain; and 
inasmuch as the kind of distortion at once calls to mind the 
manner in which the force acts, it is not essential for our pur- 
pose that the distinction should always be made. 

A Tensile Stress, or Pull, is a force which tends to elongate 
a piece, and produces a strain of extension, or tensile strain, 

A Compressive Stress, or Push, tends to shorten the piece, 
and produces a compressive strain. 

Transvere Stress acts transversely to the piece, tending to 
bend it, and produces a bending strain. But as a compressive 
stress sometimes causes bending, we call the former a transverse 
strain, for it thus indicates the character of the stress which 
produces it. Beams are generally subjected to transverse 
strains. 

Torsive Stress causes a twisting of the body by acting tan- 
gentially, and produces a torsive strain. 

Longitudinal Shearing- Stress, sometimes called a detru- 

sive strain, acts longitudinally in a fibrous body, tending to 

draw one part of a solid substance over another part of it ; as, 

for instance, in attempting to draw the piece 

A B, Fig. 1, which has a shoulder, through the 

mortise O, the part forming the shoulder will be 

forced longitudinally off from the body of the 

FlG ' 1 piece, so that the remaining part may be drawn 

through. (See also Fig. 31.) 

Transverse Shearing Stress is a force which acts trans- 
versely, tending to force one part of a solid body over the adja- 




INTRODUCTION. 6 

sent pari. It acts like a pair of shears. It is the stress which 
would break a tenon from the body of a beam, by acting per- 
pendicular to the side of the beam and close to the tenon. It 
is the stress which shears large bars of iron transversely, so 
often seen in machine-shops. The applied and resisting forces 
act in parallel planes, which are very near each other. 

Splitting Stress, as when the forces act normally like a 
wedge, tending to split the piece. 

3. THE EFFECT OF THESE STRESSES IS TWOFOLD : 1st. 

Within certain limits they only produce change of form ; and, 
2d, if they are sufficiently great they will produce rupture, or 
separation of the parts ; and these two conditions give rise to 
two general problems under the resistance of materials, the 
former of which we shall call the problem of Elastic Resist- 
ance ; the latter, Ultimate Resistance, or Resistance to Rup- 
ture. 

4. general principl.es of elastic resistances. — 

To determine the laws of elasticity we must resort to experi- 
ment. Bars or rods of different materials have been subjected 
to different strains, and their effects carefully noted. 

From such experiments, made on a great variety of materials, 
and with apparatus which enabled the experimenter to observe 
very minute changes, it has been found that, whatever be the 
physical structure of the materials, whether fibrous or granular, 
they possess certain general properties, among which are the 
following : — 

1st. That all bodies are elastic, and within very small limits 
they may be considered perfectly elastic ; i. e., if the particles 
of a body be displaced any amount within these limits they 
will, when the displacing force is removed, return to the same 
position in the mass that they occupied before the displace- 
ment. This limit is called the limit of perfect elasticity r 

* Mr. Hodgkinson made some experiments to prove that all bodies are non- 
elastic. (See Civil Eng. and Arch. Jour., vol. vi., p. 354.) He found that the 
limits of perfect elasticity were exceedingly small, and inferred that if oui 



4 THE RESISTANCE OF MATERIALS. 

2d. The amount of displacement within the elastic limit is 
directly proportional to the force which produces it. It follows 
from this, that in any prismatic bar the force which produces 
compression or extension, divided by the amount of extension 
or compression, will be a constant quantity. 

3d. If the displacement be carried a little beyond this limit 
the particles will not return to their former position when the 
displacing force is removed, but a part or all of the displace- 
ment will be permanent. This Mr. Hodgkinson called a set, a 
term which is now used by all writers upon this subject. 

4th. The amount of displacement is not exactly, but nearly, 
proportional to the applied force considerably beyond the elas- 
tic limit. 

5th. Great strains, producing great sets, impair the elasticity. 

5. COEFFICIENT (OR MODULUS*) OF ELASTICITY. 

If a prismatic bar, whose section and length are 
unity, be compressed or elongated any amount 
within the elastic limit, the quotient obtained by 
dividing the force which produces the displacement 
by the amount of compression or extension is called 
the Coefficient of Elasticity. This we call E. 
Let K = section of a prismatic bar (See Fig. 2), 
P I = its length, 

Fig. 2. 

powers of observation were perfect in kind and infinite in degree, we should 
find that no body was perfectly elastic even for the smallest amount of dis- 
placement. And although more recent experiments have indicated the same 
result in cast-iron, yet the most delicate experiments have failed to thoroughly 
establish it. I have, therefore, accepted the principle of perfect elasticity, 
which, for the purposes of this work, is practically, if not theoretically, cor- 
rect. It does not appear from Mr. Hodgkinson' s report how soon the effect 
was observed after the strain was removed. If he had allowed considerable 
time the set might have disappeared, as it is evident that it takes time for the 
displaced particles to return to their original position. 

* The terms coefficient and modulus are used indiscriminately for the con- 
stants which enter equations in the discussion of physical problems, and are 
sometimes called physical constants. The modulus of elasticity, as used by most 
writers on Analytical Mechanics, is the ratio of the force of restitution to 




INTRODUCTION. 

and a == the elongat. on or compression caused by a force, P, 
which is applied longitudinally. Then 

P 

-=. = force on a unit of section, and 

XL 

- = the elongation or compression for a unit of length. 
c 

Hence, from the definition given above, we have 

E -K^l=K* (1) 

From this equation E may be easily found. It will here- 
after be shown that the coefricent is not exactly, but is nearly 
the same for compression as for tension. 
For values of E, see Appendix III. 

6. PROOFS OF THE LAWS GIVEN IN ARTICLE FOUR. — 

Article 5 has preceded these proofs, so as to show how the re- 
sults of experiments may be reduced by equation (1). The 1st 
and 2d laws seem first to have been proved by S. Gravesend, 
since which they have been confirmed by numerous experiment- 
ers. One of the most extensive and reliable series of experi- 
ments upon various substances for engineering purposes is given 
in " The Report of Her Majesty's Commissioners, made under 
the direction of Mr. Eaton Hodgkinson." The results of his 
experiments are published in the Reports of the British Associ- 
ation, and in the 5th volume of the Proceedings of the Manches- 
ter Literary and Philosophical Society, from which extracts 
have been made and to which we shall have occasion to refer. 
The experiments were made not only to prove these laws but 
several others, principally relating to transverse strength. 

Barlow made many experiments, the results of which are 
given in his valuable work on the " Strength of Materials." 

that of compression. It relates to the impact of bodies, and, as thus defined, 
depends upon the set. But the coefficient of elasticity depends neither upon 
impact nor set. Another term should therefore be used, or else a distinction 
should be made between the terms coefficient and modulus, so that the former 
shall apply to small displacements, and the latter to the relative force of resti- 
tution. For this reason I have used the former in this work, and avoided the 
latter when applied to elasticity. 



6 THE RESISTANCE OF MATERIALS. 

The series of experiments on iron which had been commenced 
and so ably conducted by Mr. Hodgkinson were continued by 
Mr. Fairbairn, The latter confined his experiments mostly to 
transverse strength, the results of which are given in his valu- 
able work on " Cast and Wrought Iron." A valuable set of 
experiments has been made in France at " le Conservatoire des 
Arts et Metiers." * 

In this country several very valuable sets of experiments have 
been made, among the most important of which are the experi- 
ments of the Sub-Committee of the Franklin Institute, the re- 
sults of which are published in the 19th and 20th volumes of 
the Journal of that Society, commencing on the 73d page of the 
former volume. The experiments were made upon boiler iron, 
but they developed many properties common to all wrought 
iron. They were conducted with great care and scientific skill. 
The report gives a description of the testing machine; the 
manner of determining its friction and elasticity ; the modifica- 
tions for use in high temperature ; the manner of determining 
the latent and specific heats of iron ; and the strength of differ- 
ent metals under a variety of circumstances. 

Another very valuable set of experiments was made by Cap- 
tain T. J. Rodman and Major W. Wade, upon " Metals for 
Cannon, under the direction of the United States Ordnance 
Department," and published by order of the Secretary of War. 

Numerous other experiments of a limited character have been 
made, too many of which have been lost to science because they 
were not reported to scientific journals, and many others were 
of too rude a character to be very valuable 

The results of these experiments will form the basis of oui 
theories and analysis. 

* See u Morirts Resistance des Materiaux," p. 126. 



liXPERIMENTS ON WROUGHT IRON. 



CHAPTEE I. 



TENSION*. 



7. elastic resistance. — We will first consider the elas- 
tic resistance due to tension, or, as it is sometimes called, a pull, 
or elongating force. 

EXPERIMENTS ON WROUGHT IRON. 

Experiments for determining the total elongation and permanent elongation pro 
duced by different weights acting by extension on a tie of wrought iron of 
the best quality, by Eaton Hodgkinson. 





Elongation per metre of length. 




Weight in 






Coefficient of 


lrilngra.-mmf>K per square 








centimetre. 






per square metre 


P. 


Total. A. 


Permanent. 


E. 


Kil. 


M. 


MiU. 


Kil. 


187.429 


0.000082117 




22 824 500 000 


374.930 


0.000185261 
0.000283704 




20 216 200 000 


562.406 


0.00254 


19 824 100 000 


749.456 


0.000379467 


0.0033894 


19 704 000 000 


937.430 


0.000475113 


0.0042398 


19 729 909 000 


1124.813 


0.000570792 


0.00508 


19 706 000 000 


1312.283 


0.000665647 


9.0067705 


19 714 600 000 


1499 720 


0.000760311 


0.0100S79 


19 320 300 000 


1687.219 


0.000873265 


0.0330283 


19 320 700 000' 


1874. 645 


0.001012911 


0.0829955 


18 398 100 000 


2063.580 


0.001283361 


0.2616950 


16 079 200 000 


2249.627 


0.002227205 
0.004287185 






2403.653 


3.0709900 


5 606 590 000 


2624.564 


0.009156490 


8.4690700 


2 866 380 000 




0.009950970 


8.5748700 




2812.033 


0.010492805 


9.1023600 


2 681 520 000 


Repeated after 1 hour. 


0.011750313 






II II g u 


0.011858889 
0.011933837 
0.011942168 
0.011958835 
0.011967149 
0.012027114 
0.012027014 






" " 3 " 






» ii 4 u 






ii ii 5 a 




II II g u 






ll ii 7 ii 






ii ii 8 ii 







THE RESISTANCE OF MATERIALS. 
EXPERIMENTS ON WROUGHT IRON.— Continued. 



Weight in 
kilogram mop, per square 


Elongation per metre of length. 


Coefficient of 
elasticity 


centimetre. 
P. 


Total. A. 


Permanent. 


per square metre. 
E. 


Kil. 
Repeated after 9 hours. 
" " 10 " 
2999.500 
2999.500 


M. 

0.012027114 

0.012027114 

0.017888263 

0.019478898 

0.01984831 

0.02022006 

0.02148590 

0.02169401 

0.02170242 

0.02170242 

0.02477441 

0.02514184 

0.02522512 

0.03493542 

0.03519357 

0.03520190 


Mill. 


Kil. 


16.5146*" 


'i 676 820 666 




18.4212 
18.8886 
19.7954 




3i86.973 


'i 483 290 666 








3374] 440 


22.0119 

22.7087 


'i 362 626 666 


3561.900 


32.8201 


1 019 580 000 








3745.361 







This table is given in French units because it was more con- 
venient.* 

8. the results of these experiments may be re- 
presented graphically by taking, as in Fig. 3, the total elonga- 
tions or the permanent elongations for abscissas and the weights 
for ordinates. 

* To reduce the French measures to English we have the following rela 
tions: — 

Linear Measure. 

3.2808992 feet = 1 metre. 
0.328089 feet = 1 centimetre. 
0.0032808 feet = 1 millimetre. 
0.0393696 in. = 1 millimetre. 

Weight. 

2.20462 lbs. avoird. = 1 kilogramme. 

1422.28 lbs. pr. sq. in. = 1 kilog. to the sq. millimetre. 

0.00142228 lbs. sq. in. = 1 kilog. pr. sq. metre. 

Hence to reduce the above quantities to English units, multiply the numbers 
in the first column by 14.2228 to reduce them to pounds avoirdupois per square 
inch ; those in. the second column by 3.28089+ to reduce them to feet; the 
third by 0.03936+ to reduce them to inches ; and the fourth by 0.00142228 to 
reduce them to pounds per square inch. 



TENSION. 



9 



2400 



When the construction is made on a large scale it makes the 
results of the experiments very evident. 

An p-jrarniTifl.- 2GQW 



examina- 
tion of Fig. 3 
shows : — 

1st. That to a 
load of 1499.72 
kil. pr. square 
centimetre, the 2000 
total elongations 
are practically 
proportional to 120a 
the loads ; 

2d. That with- 
in the same lim- 
its the perma- 
nent elongations 
are nearly pro- 



800 



400 











Got 




~^T?tfi 












^c 


^ 


rg^Z 




pfi 1. 








/ 


y^ 


















(f 

























































































































on*- 004: 



•016 

Fig. 



•028 



•036 



portional to the loads, and that they are exceedingly small ; 

3d. That beyond the load of 14997 kil. to 22.00 kil. per 
square millimetre, the total and permanent elongations increase 
very rapidly and more than proportional to the loads ; 

4th. That near and beyond 22.49 kil. per square millimetre, 
the elongations become sensibly proportional to the increments 
of the loads, but in a much greater ratio than that which corre- 
sponds to small loads. For the loads near rupture the elonga- 
tions are a little inferior to that indicated by the new proportion. 

5th. Beyond 14.99 kil. per square millimetre, the permanent 
elongations increase much more rapidly than the total elonga- 
tions. We also observe that the permanent elongations increase 
with the duration of the load, although very slowly. The latter 
property will be more particularly noticed hereafter. 

6th. Finally, the values — of the loads per square metre to 

A, 

the elongation per metre, and which is called the coefficient of 
elasticity, is sensibly constant when the elongations are nearly 
proportional to the loads ; and that the mean value is 
E= 19,816,440,000 kil. per square metre ; 
= 28,285,000 lbs. per square inch. 
The first value of E, in the table, is much larger, and may 



10 THE RESISTANCE OF MATERIALS. 

have resulted from an erroneous measurement of the exceed 
ingly small total elongation. From the experiments made on 
another bar, Hodgkinson found 

E == 19,359,458,500 kil. per sq. metre ; 
= 27,700,000 pounds pr. square inch ; 
which is but little less than the preceding. 

Mr. Hodgkinson infers from these experiments that the small- 
est strains cause a permanent elongation. Bnt Morin for- 
cibly remarks * that none of these experimenters appear to have 
verified whether time, after the strains are removed, will not 
cause the permanent elongations to disappear. Also that the 
deflections of the machine cannot be wholly eliminated, and 
hence appear to increase the true result. In practice such small 
permanent elongations may be omitted. 

The preceding example has, for a long time, been given to 
show the law of relation between the applied force and the 
total and permanent elongations ; bnt we should not expect to 
find exactly the same results for all kinds of iron. Even wrought 
iron has such a variety of qualities, depending npon the ore of 
which it is made, and the process of manufacture, that it cannot 
be expected that the above results will always be applicable to 
it. Only a wide range of experiments will determine how far 
they may generally be relied upon. 

It is found, however, that the general results of extension, 
of set, of increased elongation with the duration of the stress 
within certain limits, and of the increase of set with the in- 
crease of load, are true of all kinds of iron. 

EXPERIMENTS UPON CAST IRON. 
9. THE FOLLOWING EXPERIMENTS UPON CAST-IRON show 

that the numerical relation between the applied force and the 
extension is somewhat different from the preceding. The expe- 
riments were made under the supervision of Captain T. J. Bod- 
man : — f 

" The specimens had collars left on them at a distance of thirty-five inches 

* Morin's Resistance des Materiaux, p. 10. 

f Experiments on Metals for Cannon, by Capt. T. J. Rodman, p. 157. 

For a full description of the testing 1 apparatus, with diagrams, see Major 
Wade's Report on the Strength of Materials for Cannon, pp. 305-315. The 
machine consists principally of a very substantial frame vnd levers resting or 
knife edges. 



TENSION. 



11 



apart, the space between the collars being accurately turned throughout to a 
uniform diameter. 

" The space between the collars was surrounded by a cast-iron sheath, eight- 
tenths of an inch less in length than the distance between the collars ; it was 
put on in halves and held in position by bands, and was of sufficient interior 
diameter to move freely on the specimen. 

"When in position, the lower end of the sheath rested on the lower collar 
of the specimen, the space between its upper end and the upper collar being 
supplied with and accurately measured by a graduated scale tapered 0. 01 of an 
inch to one inch. 

' ' The upper end of the sheath was mounted with a vernier, and the scale 
was graduated to the tenth of an inch. 

" This afforded means of measuring the changes of distance between the 
collars to the ten- thousandth part of an inch, and these readings divided by the 
distance between the collars gave the extension per inch in length as recorded 
in the following table : — 

TABLE 
Showing the extension and 'permanent setter inch in length caused by the under- 
mentioned weights, per square inch of section, acting upon a solid cylinder 35 
inches long and 1.366 inches diameter. (Cast at the West Point Foundry in 
1857.) 



Weight per 

square inch of 

section. 


Extension per inch of 
length. 


Permanent set per inch 
in length. 


Coefficient of 
elasticity. 


P. 


A. 




E. 


lbs. 


in. 


in. 




1,000 • 


0.0000611 


0. 


16,366,612 


2,000 


0.0000794 


0. 


25,189,168 


3,000 


0.0001089 


0. 


27,548,209 


4,000 


0.0001771 


0. 


22,586,674 


5,000 


0.0002129 


0. 


23,489,901 


6,000 


0.0002700 


0.0000014 


22,222,222 


7,000 


0.0003328 


0.0000029 


21,033,653 


8,000 


0.0003986 


0.0000043 


20.070,245 


9,000 


0.0004557 


0.0000071 


19,749,835 


10,000 


0.0005100 


0.0000109 


19.607,843 


11,000 


0.0005500 


0.0000157 


20^00,000 


12,000 


0.0006414 


0.0000257 


18,693,486 


13,000 


0.0007100 


0.0000300 


18,309,859 


14,000 


0.0007700 


0.0000357 


18,181,181 


15,000 


0.0008557 


0.0000477 


17,529,507 


16,000 


0.0009243 


0.0000529 


17,310,397 


17,000 


0.0010014 


0.0000643 


16,977,231 


18,000 


0.0010900 


0.0001014 


16,537,614 


19,000 


0.0012271 


0.0001471 


15,483,660 


20.000 


0.0013586 


0.0002014 


14,721,109 


21,000 


0.0015386 


0.0002900 


13,648,771 


22,000 


0.0017043 


0.0003986 


12,908,523 


23,000 


0.0019529 


0.0005529 


11,265,246 


24,000 


0.0022786 


0.0007529 


10,532,344 


25,000 


0.0026037 


0.0010843 


9,601,720 


26,000 


0.0032186 




8,078,046 



12 



THE RESISTANCE OF MATERIALS. 



10. FIGURE 4 IS A GRAPHICAL REPRESENTATION OF 

the above table, constructed in the same way as Figure 3. 
Experiments were made upon many other pieces, from which 
I have selected four, and called them A, B, O, and D 9 a gra- 
phical representation of which is shown in Figure 5. The right 
hand lines represent extensions, the left hand sets. 

















<$> 






$k 
























/ 












/ 














/ 
















Fig. 4. 



Fig. 5. 



A was from an inner specimen of a Fort Pitt gun, No. 335, 
and the others from different cylinders which were cast for the 
purpose of testing the iron. 

From these we observe : — 

1st. That for small elongations the ratio of the stresses to the 
elongations is nearly constant. 

2d. There does not appear to be a sudden change of the rate 
of increase, as in Mr. Hodgkinson's example, but the ratio gra- 
dually increases as the strains increase. 

3d. The sets at first are invisible, but they increase rapidly 
as the strains approach the breaking limit. 

It appears paradoxical that the first and second experiments 
in the preceding table should give a less coefficient than the 
third, but the same result was observed in several cases. 

11. THE FOLLOWING TABLES ARE THE RESULTS OF 
SOME EXPERIMENTS MADE BY MR. HODGKINSON :— 



TENSION. 



13 



Direct longitudinal extension of round rods of cast iron, fifty feet long. 





CO 


o 


Weights per square inch laid on, 


? §1 


| 




«H C 


* A 


with their corresponding ex- 


§§ 




O 0) 


£ o 




0> p o 


ll 


NAME OP IEON. 




d£ 








0? 


g M 




A -8° 

ag-a 


cl-S 




Weights. 
P. 


Extension. 
A. 


Sets. 










lbs. 


in. 


in. 


lbs. 


in. 


Low Moor, No. 2 


2 


1.058 


2,117 

6,352 

10,586 

14,821 


0.0950 
0.3115 
0.5640 
0.9147 


0.00345 
0.0250 
0.4425 
0.12775 


16,408 


1.085 


Blaneavon Iron, No. 2.. 


2 


1.0685 


2,096 

6,289 

10.482 

13J627 


0.0942 
0.3065 
0.5770 

0.8370 


0.00268 
0.01675 
0-0575 
0.11475 


14,675 


0.9325 


G-artsherrie Iron, No. 2. 


2 


1.062 


2,109 

6,328 

10,547 

14,766 


0.0922 
0.3117 
0.5862 
0.9452 


0.001 + 
0.01450 
0.0475 
0.11352 


16,951 


1.167 



In these experiments the ratio of the extensions is somewhat 
greater than that of the weights. The value of £J, as computed 
for the first weights which are given, and the corresponding 
extensions, is a little more than 13,000,000 pounds per square 
inch. 











liliSS 


Extension of cast-iron rods, ten feet long and one inch square. 


a !rasi 










*»*M% 


Weights 
P. 


Extensions. 

Ae. 


Sets. 


p. 
aT. 


Erroi 
of t 
when 
putecl 
mula 
A e - 


lbs. 


in. 


in. 






1053.77 


.0090 


00022 


117086 


L. 


1580.65 


.0137 


000545 


115131 


i 


2167.54 


.0186 


00107 


113309 


~~h 


3161.31 


.0287 ■ 


.00175 


110150 


+ eio 


4215.08 


.0391 


.00265 


107803 • 


+ 2 br 


5262.85 


.0500 


.00372 


105377 


tt 


6322.62 


.0613 


.00517 


103142 


7376.39 


.0734 


.00664 


100496 


+ 7^6 


8430.16 


.0859 


.00844 


98139 


+-ihr 


9483.94 


.0995 


.01062 


95316 


+t£* 


10537.71 


.1136 


.01306 


92762 


+ t¥t 


11591.48 


.1283 


.01609 


90347 


-eh: 


12645.25 


.1448 


.02097 


87329 


-tS 


13699.83 


.1668 




82133 


+?h 


14793.10 


.1859 


.02410 


79576 


-& 



14 THE RESISTANCE OF MATERIALS. 

Let P — the elongating force and 

A e = the total elongation in inches due to P. 
Then Hodgkinson found, from an examination of the table, 
that the empirical formula 

P = 116117^ - 201905a* 

represented the results more nearly than equation (1). This for- 
mula reduced to an equivalent one for I in inches (observing 
that the bar was 10 feet long), becomes 

P = 13,934,000 j - 2,907,432,000^ 

Although this equation gives the elongations for a greater range 
of strains than equation (1) for this particular case, yet the 
law represented by it is more complicated, and hence would 
make the discussions under it more difficult, without yielding 
any corresponding advantage. It is the equation of a parabola 
in which P is the abscissa and A e the ordinate. 

We also see that when the elongations are very small, the 

A 2 

quantity -^ will be very small, and the second term may be 

omitted in comparison with the first, in which case it will be re- 
duced to equation (1). The coefficient in the first term is the 
coefficient of elasticity, hence it is nearly 14,000,000 lbs. for 
extension. 

MALLEABLE LRON. 

IS. according to barlows experiments malleable 
iron may be elongated 10 1 00 of its length without endangering 
its elasticity.* To ascertain this, the strains were removed 
from time to time, and it was found that the index returned to 
zero for all strains less than 9 or 10 tons. The mean extension 
per ton (of 2,240 lbs.) per square inch, for four experiments, was 
0.00009565 of its original length. Hence the mean value of 
the coefficient of elasticity is 
j> 2940 

^-T = a00009565 = 23 ' 418 '° 00lbs - 



* Journal Frank Inst., vol. xvi., 2d Series, p. 126. 



TENSION. 15 

ELASTICITY OF WOOD. 
13, EXPERIMENTS BY MESSRS. CHEVANDIER AND WER- 

theim. — The following are some of the results of the recent 
experiments of Messrs. Chevandier and "Wertheim on the resist- 
ance of wood. These experimenters have drawn the following 
principal conclusions : — 

1. The density of wood appears to vary very little with age. 

2. The coefficient of elasticity diminishes, on the contrary, be- 

yond a certain age ; it depends, likewise, upon the dry- 
ness and the exposure of the soil to the sun in which the 
trees have grown ; thus the trees grown in the northern 
exposures, north-eastern, north-western, and in dry soils, 
have always so much the higher coefficient as these two 
conditions are united, whereas the trees grown in muddy 
soils present lower coefficients. 

3. Age and exposure influence cohesion. 

4. The coefficient of elasticity is affected by the soil in which 

the tree grows. 

5. Trees cut in full sap, and those cut before the sap, have not 

presented any sensible differences in relation to elasticity. 

6. The thickness of the woody layers of the wood appeared to 

have some influence on the value of the coefficient of 
elasticity only for fir, which was greater as the layers were 
thinner. 

7. In wood there is not, properly speaking, any limit of elas- 

ticity, as every elongation produces a set. 
It follows from this circumstance that there is no limit of 
elasticity for the woods experimented upon by Messrs. Chevan- 
dier and Wertheim ; but, in order to make the results of their 
experiments agree with those of their predecessors, the authors 
have given for the value of the limit of elasticity the load under 
which it produces only a very small permanent elongation ; the 
limit which they indicate in the following table for loads, under 
which the elasticity of wood commences to change, corresponds 
to a permanent elongation of 0.00005 of its original length. 



16 



THE RESISTANCE OF MATERIALS. 



Table containing the Mean Results of the Experiments oi 
Messrs. Chevandier and Wertheim. 



Species. 


1 

S3 

<v 
ft 


Coefficient of elas- 
ticity E referred to 
the square milli- 
metre. 


Limit of elasticity, 
or load per square 
millimetre, corres- 
ponding to that li- 
mit. 


Cohesion, or load 
per square milli- 
metre capable of 
producing rupture. 


M 

§,a 
.a| 

e3 ft 
> 


Locust 


0.717 
0.493 
0.756 
0.812 
0.823 
0.808 
0.872 
0.559 
0.723 
0.692 
0.697 
0.601 
0.602 
0.674 
0.477 


Kilogr. 

1261.9 
1113.2 

1085.3 

997.2 

980.4 

977.8 

921.8 

564.1 

1165.3 

1163.8 

1121.4 

1108.1 

1075.9 

1021.4 

517.2 


Kilogr. 

3.188 
2.153 

1.282 

1.617 

2.317 
<( 

2.349 
1.633 
1.842 
1.139 
1.246 
1.121 
1.035 
1.068 
1.007 


Kilogr. 

7.93 
4.18 
2.99 
4.30 
3.57 
6.49 
5.66 
2.48 
6.99 
6.16 
6.78 
4.54 
7.20 
3.58 
1.97 


$£? 


Fir 


'S §j 


Yoke Elm 


23! 


Birch 


&l a 


Beech 


2*8 


Oak from pedunculate acorn. 

" " sessile acorn 

White Pine 


•43 £ 

02 . J3 


Elm 


00 & 

+3 O " 


Sycamore ... 


s °~ « 


Ash 




Alder 


Aspen 


g 1 "" 1 © 


Maple 


s«2 

© SO 


Poplar 


A 8°°- 







141. ELASTICITY OF WOO», TANGENTIALLY AND RADI- 
ALLY. — The same observers have also determined the coefficient 
of elasticity and the cohesion of wood in the direction of the 
radius and in the direction of the tangent to the woody layers. 

An examination of the following Table shows that the resist- 
ance in the direction of the radius is always greater than the 
resistance in the direction of the tangent to the woody layers ; 
the relation between the coefficients of elasticity in the two 
cases varying nearly from 3 to 1.15. 



TENSION. 



17 



Mean Results of the Experiments of Messrs, Chevandier 
and Wertheim. 





In the Direction of Radius. 


In the Direction of the Tan- 
gent to the Layers. 


SPECIES. 


Coefficient of 
Elasticity, E, per 
square millime- 
tre. 


Cohesion, or 
load, per square 
millimetre, capa- 
ble of producing 
rupture. 


Cohesion, or 

Coefficient of load, per square 

Elasticity, E, per millimetre, capa- 

square millime- ble of producing 

tre. rupture. 


Yoke Elm 


Kilogr. 

208.4 

134.9 

157.1 

188.3 

81.1 

269.7 

111.3 

121.6 

94.5 

97.7 

170c 3 


Kilogr. 

1.007 
0.522 
0.716 
0.582 
0.823 
0.885 
0.218 
0.345 
0.220 
0.256 


Kilogr. 

103.4 

80.5 

72.7 

129.8 

155.2 

159.3 

102.0 

63.4 

34.1 

28.6 

152.2 


Kilogr. 

0.608 


Sycamore 

Maple 


0.610 
0.371 


Oak 


0.406 


Birch 


1.063 


Beech.. 

Ash 


0.752 
0.408 


Elm 


0.366 


Fir 


0.297 


Pine 


0.196 


Locust 


1.231 













The highest coefficient of elasticity in this table is for beech, and this is less 
than 400,000 pounds per square inch. 



15. remark. — The value of E, which is used in practice, is 
not the coefficient oft. perfect elasticity, but it is that value which 
is nearly constant for small strains. In determining it, no ac- 
count is made of the set. If the total elongations were propor- 
tional to the stresses which produce them, we would use the 
value of E found by them, even if the permanent equalled the 
total elongations. But in practice the permanent elongations 
will be small compared with the total for small stresses. 

APPLICATIONS. 

16. TO FIND THE ELONGATION OF A PRISMATIC RAR 
SUBJECTED TO A LONGITUDINAL STRAIN WHICH IS WITH- 
IN THE ELASTIC LIMITS. 



From (1) we have 



A = 



PI 

EK 



(2) 



which is the required formula. 

2 



18 



THE RESISTANCE OF MATERIALS. 



Also from (1) we have 



EK 



(3) 




Fig. 6. 



Equations (1), (2), and (3) are equally applicable 
to compressive strains, as will hereafter be shown. 
If in (3) we make K —\ and a — Z we shall have 
P — E\ hence, the coefficient of elasticity may be 
defined to be a force which will elongate a bar whose 
section is unity, to double its original length, pro- 
vided the elasticity of the material does not change. 
But there is no material, not even a perfectly elastic 
body — as air and other gases — whose coefficient of 
elasticity will not change for a perceptible change of volume. 
The material may not lose its elasticity, but equation (1) only 
measures it for small displacements. To illustrate further, let 
it be observed that, according to Mariotte's law, the volumes of 
a gas are inversely proportional to the compressive (or exten- 
sive) forces ; double the force producing a compression to half 
the volume ; four times the force, to one-fourth the volume, and 
so on, the compressions being & fractional part of the original 
volume ; but in equation (2), a is a linear quantity, so that if 
one pound produces an extension (or compression) of one inch, 
two pounds would produce an extension of two inches, and 



so on. 



Examples. — 1. If the coefficient of elasticity of iron be 25,000,000 lbs., what 
must be the section of an iron bar 60 feet long, so that a weight of 5,000 lbs. 
shall elongate it £ an inch. 

PI • 

From (1) we obtain K = ~ which by substitution becomes 

Mi\ 

_ 5,000.12x60 „... 

K = xjatojMZi = a288 6qnare mches - 

2. What weight will a brass wire sustain, whose diameter is 1 inch, coeffi« 
tient of elasiicity is 14,000,000 lbs., so as to elongate it -^ of its length ? 

Ans. 13,744.5 lbs. 



TENSION. 



19 



/ 



EK 



ax _ EK 




17. — REQUIRED THE ELONGATION (OR COMPRESSION) OF 
A PRISMATIC BAR WHEN ITS WEIGHT IS CONSIDERED. 

Let I = the whole length of the bar before elon- 
gation or compression, 

x = variable distance = Ah, 

dx = be = an element of length, 

w = weight of a unit of length of the bar, 

W = weight of the bar, and 

P x — the weight sustained by the bar. 
Then (I — x)w -\r JP 1 = JP = the strain on any 
section, as be. 

Hence, from equation (2), we have 




Fig. 7. 



(*) 



the total length will become, 



I ± X 



-D*^] 



If P 



0, A = 



wV 



Wl 



(5) 



9 tpt^i or tne tota ^ elongation is one- 
half of what it would be if a weight equal to the whole weight 
of the bar were concentrated at the lower end. 

Required the Elongation (ok Compression) of a Cone in 
a Vertical Position, caused by its own Weight when it is 

SUSPENDED AT ITS BASE (OR RESTS ON ITS Base). 

Take the origin at the apex before mm//M////MiiMMm0/////////ffL 
extension, Fig. 8, and 
let iff — any section, 

K = the upper section, 



the length or altitude of 



I 

the cone, 
x = the length or altitude of 

any portion of the cone, 

and 
8 == the weight of a unit of volume. 




Fig. 8. 



Then, because the bases of similar cones are as the squares 

x 9 
of their altitudes, K = iT -=- 



20 THE KE6ISTANCE OF MATERIALS. 

The volume of the cone whose altitude is x 

=f*Kdx =./V.-£*> = iKj^ 

and the weight of the same part 
= * 8 K % -J- 



8 JT 
(from equation (2)) a = / -J — jr-as' 



■/ 



■££".-£ 



<#a? = 



1_S? 
6 JE* 



from which it appears that the total elongation is independent 
of the transverse section, and varies as the square of the 
length. 

1 8. the work of elongation. — If P be the force 
which does the work, and x the space over which it works, then 
the general expression for the work is 



f X Pdx, 



(6) 



To apply this to the elongation of a prism, substitute P from 
Eq. (3) in (6), and make dx — d\ and we have 

u= /* X *£A=™=tf* (7) 




I 21 

which is the same result that we would have found by suppos- 
ing that P was put on by increments, increasing the load gradu- 
ally from zero to P. 

Example. — If the coefficient of elasticity of wrought iron be 28,000,000 lbs., 
and is expanded 0.00000698 of its length for one degree F., bow much work is 
done upon a prismatic bar whose section is one inch, and length 30 feet, by a 
change of 20 degrees of temperature ? 

Walls of buildings which were sprung outward have been drawn into an erect 
position by heating and cooling bars of iron. Several rods were passed through 
the building, and extending from wall to wall, were drawn tight by means of 
the nuts. Then a part of them were heated, thus elongating them, and the 
nuts tightened ; after which they were allowed to cool, and the contraction 
which resulted drew the walls together. Then the other rods were treated in 
a similar manner, and so on alternately. 



TENSION. 



21 



19. VERTICAL OSCILLATIONS.— If a bar Aa, Fig. 9, 
with a weight, P, suspended from its lower end, be pressed down 
by the hand, or by an additional weight from a to b, and the addi- 
tional force be suddenly removed, the end of the bar on returning 
will not stop at a, but will move to some point above, as c, a dis- 
tance ac = ab. From a principle in Mechanics, viz. , that the living 
force equals twice the work, we are enabled to determine all the 
circumstances of the oscillation when the weight of the bar is 
neglected. The weight P elongates the bar so that its lower ex- 
tremity is at a, at which point we will take the origin of co-or- 
dinates. 




Fig. 9. 



Let A = ab = the elongation caused by the additional force, 

x = ad = any variable distance from the origin, 

v = the velocity at any point, as d, and 
M = the mass of the weight P. 
If the weight of the rod be very small compared with P, the vis viva is 



Mv 2 



— v 2 very nearly. 



The work for an elongation equal to A, is by Eq. (7), 



EK 
21 ' 
EK 



The work for an elongation equal to x, is by Eq. (7), -^y ar 2 



-sj 



PI 

gEK 



I. 



dx 



v/" 



= J?i sin- 1 *]* -jlJjl 

V gEK AJo 2 V gEK 



for half an oscillation ; and the time for a whole oscillation is 

*=Tvffl5="\/7 • (8 ' 

hence the oscillations will be isochronous. 

It is evident that by applying and removing the force at regular intervals, the 
amplitude of the oscillations may be increased and possibly produce rupture. 
In this way the Broughton suspension bridge was broken.* 

As a second example, take the case in which P is applied suddenly to the end 
of the rod. It is evident that the total elongation will be greater than A, — the 
permanent elongation. For the fundamental equation we may use another 



* Mr. E. Hodgkinson, in the 4th volume. of the Manchester PhilosopJiical 
Transactions, gives the circumstances of the failure, from this cause, of the 
suspension bridge at Broughton, near Manchester, England. And M. Navier, in 
his theory of suspension bridges (Ponts Smpendus, Paris, 1823), states that 
the duration of the oscillation of chain bridges may be nearly six seconds. 



22 THE RESISTANCE OF MATERIALS. 

principle in Mechanics, which might have been used in the preceding problem, 

viz. , that the mass multiplied by the acceleration equals the resultant moving 

EKx 
force. The resisting force for an elongation x is — — - (See Eq. (3) ), and the 



,,<Re „ EK 



t^x dx 



t rm x ax 



^. * versin" 1 ^ 
g EK pi 



If x = A, v = v gl, 
x = 2)., v=0, 
x = 0, ' v = 0. 
Hence, the amplitude is twice the permanent elongation. If x = 2X we have 

/~Pl l~T 
t = 7i / —jpjr = n / Investigations of this kind give rise to a divi- 
sion of the subject called Besiliance of Prisms. 

The investigations are interesting, but the results are of little use beyond 
those which have already been indicated. From the last problem we see that a 
weight suddenly applied produces twice the strain that it would if applied 
gradually. 

As additional exercises for the student, I suggest the following : Suppose the 
weight be applied with an initial velocity. Suppose a weight P is attached to 
one end, and the weight P' is placed suddenly upon it ; or it falls upon it. 
To find the velocity at any point in terms of £, — also A in terms of t. 

If a weight Wis, suspended at the end, and another weight TFi falls from a 
height h, giving rise to a velocity v, we have for the common velocity of the 

bodies after impact, if both are non-elastic, V= -^ — ^?, and the vis viva of 
both will be 



WiV EK 

MV " = f(w^ w wMeih equaIs _ r* 



i= w - 



12 hi 
>\/ EK 



v(Wi+wy 

This is only an approximate value, for the inertia of the wire is neglected. 

SO. viscosity of solids. — Experiments show that the prin- 
ciple of equal amplitudes, referred to in the preceding article, is 
not realized in practice. This is more easily observed in trans- 
verse vibrations. The amplitudes grow rapidly less from the 
first vibration, and the diminution cannot be fully accounted for 
by the external resistance of air. Professor Thomson of Eng- 



TENSION. 23 

land has shown that there is an internal resistance which opposes 
motion among the particles of a body, and is similar to that 
resistance in fluids which opposes the movement of particles 
among themselves. He therefore called it viscosity* He 
proved : — 

1st. That there was a certain internal resistance which he 
called Viscosity, and which is independent of the elastic pro- 
perties of metals ; 

2d. That this force does not affect the co-efficient of elasti- 
city. 

The law between molecular friction and viscosity was not 
discovered. 

The viscosity was always much increased at first by the in- 
crease of weight, but it gradually decreased, and after a few 
days became as small as if a lighter weight had been applied. 
Only one experiment was made to determine the effect of con- 
tinual vibration ; and in that the viscosity was very much in- 
creased by daily vibrations for a month. 

This latter fact, if firmly established, will prove .to be highly 
important ; -for it shows that materials which are subjected to 
constant vibrations, such as the materials of suspension bridges, 
have within themselves the property of resisting more and more 
strongly the tendency to elongate from vibration. Experiments 
will be given hereafter which tend to confirm this fact, when 
the vibrations are not too frequent or too severe. 

But the true viscosity of solids has been fully proved by M. 
Tresca, a French physicist, who showed that when solids are 
subjected to a very great force, the amount of the force depend- 
ing upon the nature of the material, that the particles in the 
immediate vicinity of pressure will Jlow over each other, so as to 
resemble the flowing of molasses, or tar, or other viscous fluids. 
By applying sufficient pressure solid bodies may be made to 
flow through holes in other bodies. Thus, true viscosity 
differs entirely in its character from the property recognized by 
Professor Thomson. 

* Civ. Eng. Jour., vol. 28, p. 322. 



24 RESISTANCE OF MATERIALS. 



EESISTAISTCE TO EUPTUEE BY TENSION. 



21. modulus of strength. — Many more experiments 
have been made to determine the ultimate resistance to rupture 
by tension than there have to determine the elastic resistance. 
In the earlier experiments the former was chiefly sought, and 
more recently all who experimented upon the latter also deter- 
mined the former. 

The force which is necessary to pull asunder a prismatic bar 
whose section is one square inch, when acting in the direction 
of the axis of the bar, is called the modulus of strength. This 
we call T. It expresses the tenacity of the material, aud is 
sometimes qalled the absolute strength and sometimes modulus 
of tenacity. 

22. FORMULA FOR THE MODULUS OF STRENGTH; OT the 

force necessary to break a prismatic bar, when acted upon by 
a tensile strain. 

Let K = the section of the bar in inches ; 
T = the modulus of tenacity ; and 
P = the required force. 
It is proved by experiment that the resistance is proportional 
to the section ; hence 

P = TK. (9) 

•••^=y ( 10 ) 

From (10) T may be found. In (10) if P is not the ultimate 
resistance of the bar, then will T be the strain on a unit of sec- 
tion. 

From (9) we have 

K=^ (U) 

which will give the section. 



TENSION. 25 

The following are some of the values of T which have been 
found from experiment by the aid of Equation (10). 

Cohesive force or Tenacity 
in lbs. per square inch. 

Ash {English) 17,000 

Oak {English) 9,000 to 15,000 

Fine {pitch) 10,500 

Cast Iron* 14,800 to 16,900 

Cast Iron ( Weisbach c& Overman) 20,000 

Wrought Iron '. 50,000 to 65,000 

Steel Wire 100,000 to 120,000 

Bessemer Steel f 120,000 to 129,000 

Bessemer Steel % 72,000 to 101,000 

Bars of Crucible Steel § 70,000 to 134,000 

Chrome Steel || 115,780 to 190,680 

The most remarkable specimen of cast steel for tenacity 
which is on record was manufactured in Pittsburgh, Pa. It 
was tested at the Navy Yard at Washington, D. C, and was 
found to sustain 242,000 lbs. to the square inch ! *f 

For other values see the Appendix. 

23. A vertical prismatic bar is fixed at its upper end, and 
a weight P l is suspended at the other • what must be the upper 
section at A, Fig. 7 ' 9 so as to resist n-times all the weight below 
it, the weight of the bar being considered f 

Let ^ = the weight of a unit of volume of the bar, and the 
other notation as before. 

Then^T= nP, + niKl. 

■■■*-■& < is > 

p 

If n = 1, K = -rjrT, 5 an ^ ^ ^ ~ T, K — oo ? or no section 
1— H 

T . 

is possible, and I = -y is the corresponding length of the bar. 

* Hodgkinson, Bridges. Weale, sup., p. 25. 
f Jour. Frank. Inst. Vol. 84, p. 366. 

X Also Experiments by Wm. Fairbairn, Van JVoslrand's Eg. En. Mag., Vol 
1, p. 273. § Do. p. 1009. 

1 Report, Cat., J. B. Eads, C.E. 
1 Am. R. R. Times (Boston), Vol. 20, p. 206. 



THE KESISTANCE OP MATERIALS. 



24. bar of uniform strength. Suppose a bar is 

fixed at its upper extremity. Fig. 1© 5 and a weight P x is sus- 
pended at its lower extremity ; it is required to find the form 
of the bar so that the horizontal sections shall be proportional 
to the strains to which they are subjected — the weight of the 
bar being considered. 

Let <J = weight of an unit of volume, 
W = weight of the whole bar, 

P 
K = ~ = the section at B (Eq. (11) ), 

K x = the upper section, 
]£ z=z variable section, and 
x = variable distance from B upwards. 
Also let the sections be similar : 

Then P = P x + 8 f K dx = strain on any 

Fig. 10. sec tion, as D C. But TK is the ability to resist 
this strain : 

.-. -P x + 2 f Kdx = TK. Differentiate this, and we have 

3 Kdx = TdK 
i , dK 




or -^ dx 

T X 



— yr- which by integrating gives 
JVap. log K + G (12a) 



But for x = 0, we have K = K . * . 0= — JVap. log K = — 
Nap. log ~. Hence Eq. (12a) becomes -=, x = JVap. log J 

%x K 

or, passing to exponentials, gives e 1 



T. 



K 



.-.K=K Q e* M 
For the upper section K 



P, S® 
T 

K t and x 



(13) 



l.-.K, 



%'* (14) 



We also have 



W 



J / Kdx = 3 1 
Jo J o 



Eie^dx^P^er-l) (15) 



TENSION. 



27 



Example. What must be the upper section of a wrought-iron shaft of uni- 
form resistance 1,000 ft. long-, so that it will safely sustain its own weight and 
75, 000 lbs.?- 

Let T- 10,000 lbs., and 
h = 0.27 lbs. per cubic inch. 
Then Eq. (11) gives K = 7.5 sq. inches, and 
equation (14) gives K Y = 10.37. 

In these formulas the form of section does not appear. For 
tensile strains, the strength is practically independent of the 
form, but not so for compression. When it yields by crushing, 
the influence of form is quite perceptible, but not so much so 
as when it yields by bending under a compressive strain. The 
latter case will be considered under the head of flexure. 

25. STRAINS IN A CLOSED CYLINDER. 

If a closed cylinder is subjected to 
an internal pressure, it will tend to 
burst it by tearing it open along a 
rectilinear element, or by forcing the 
head off from the cylinder, by rup- 
turing it around the cylinder. First, 
consider the latter case. The force 
which tends to force the head off is 
the total pressure npon the head, and 
the resisting section is the cylindrical 
annulus. 

Let D = the external diameter, 
d = the internal diameter, 
jp = the pressure per square inch, and 
t = the thickness of the cylinder. 
Then \n d 2 j? = the pressure upon the head, 

i» (I) 2 — d 2 ) = the area of the cylindrical annulus, 
Jt T(D 2 — d 2 ) — the resistance of the annulus, and 
2t = D - d 
Hence, for equilibrinm, 

i*-d 2 p = i*T(D 2 - d 2 ) 
or, d 2 p = 2Tt {Z> + d) = ±T(f + dt) (16) 




Fig. 11. 



p.d 



which solved gives t = (— 1+./ 1 + 2L\ 

Equation (16) may be written as follows :■ 
d 2 j? = 4:Tt(t + d), 



.(W) 



28 THE RESISTANCE OF MATERIALS. 

and as t is generally small compared with d, we have dp = 
4 Tt nearly. 

Next consider the resistance to longitudinal rupturing. As 
it is equally liable to rupture along any rectilinear element 
suppose that the cylinder is divided by any plane which passes 
through the axis. The normal pressure upon this plane is the 
force which tends to rupture it, and for a unit of length is 

pd 
and the resisting force is 

*2Tt, 
hence, for equilibrium, 

pd = ZTt (18) 

The value of t from (18) divided by that of t from (16) gives 
the ratio — y — , and since D always exceeds d, this ratio is 

greater than 2 ; hence there is more than twice the danger of 
bursting a boiler longitudinally than there is of bursting it 
around an annulus when the material is equally strong in both 
directions. 

The last equation was established by supposing that all the 
cylindrical elements resisted equally, but in practice they do 
not ; for, on account of the elasticity of the material, they will 
be compressed in the direction of the radius, thus enlarging the 
internal diameter more than the external, and causing a corre- 
sponding increase of the tangential stress on the inner over the 
outer elements. In a thick cylindrical annulus it is necessary 
to consider this modification. 

To find the varying law of tangential strains, let D and d 
be the external and internal diameters before pressure, and 
D+z and d+y the corresponding diameters after pressure. 
Then, as a first approximation — which is near enough for prac- 
tice — suppose that the volume of the annulus is not changed, 
and we have 

fr {JD'-d 2 ) = l* (D+z)* -i*(d + yj 
or, Dz = dy nearly (19) 

But the strain upon a cylindrical filament varies as its elon 



TENSION. 29 

gation divided by its length ; see Eq. (3). Hence the strain on 
the external annulus, compared with the internal, is as 

.(D +*)-*# to '(d+y)—d m as • to %. 
*D *d D d 

which combined with (19) gives 

-™ to -y- or as d 2 to D*. or as r* to E? 
JJ d ' 

where r and i? are radii of the annnlus. 

Hence, the strain varies inversely as the square of the dis- 
tance from the axis of the cylinder. ' 

To FIND THE TOTAL RESISTANCE, let 

x — the variable distance from the axis of the cylinder, 

T — the modulus of rupture, or of strain, and 

t = the thickness of the annulus. 

Then Tdx is the strain on an element at a distance r from 

the axis of the cylinder, or otherwise upon the inner surface of 

the cylinder ; and according to the principle above stated, 

r 2 . 

T— dx is the strain on any element, and the total strain on both 

sides is 






(20) 



r + t 

If t = r, this becomes 

Tt 

which compared with Eq. (IS) shows that when the thickness 
equals the radius, the resistance is only half what it would 
be if the material were non-elastic. In (20) if t is small com- 
pared with r, it becomes 2 Tt nearly, which is the same as 
equation (IS). 

If the ends of the cylinder are capped with hemispheres, the 
stress upon an elementary annulus at the inner surface is 
2tt Trdx* Proceeding as before, and we find that the total 

* T. J. Rodman says the resistance on any elementary annulus is T-2zdx (" Exp. 
on Metal for Cannon," p. 44) ; but it appears to me that, to make his expression 
correct,' T must be the modulus at any element considered, and hence variable, 
whereas it should be constant. The strain on any elementry annulus whose 



30 



THE RESISTANCE OF MATERIALS. 



stress necessary to force the hemispherical heads off is 



2ttT- 



r + t 



(21) 



which is also the stress necessary to force asunder a sphere by 
internal pressure, when the elasticity is considered. 

If cylinders are formed by riveting together plates of iron, 
their strength will be much impaired along the riveted section. 
The condition of the riveted joint will doubtless have much 
more to do with the strength than the compressibility of tho 
material, and will hereafter be considered. 

The preceding principles are especially applicable to homo- 
geneous metals, where the thickness is considerable, such as can- 
nons and spherical shells. 



36. 



RESISTANCE OF GliASS GLOBES TO INTERNAL PRES- 
SURE. 



EXPERIMENTS OF WILLIAM FAIRBAIRN. 



Description of the glass. 


Diameter in 
inches. 


Thickness in 
inches. 


Bursting pres- 
sure in lbs. per 
square inch. 


Bursting pres- 
sure in lbs. per 
square inch of 
section. 


Flint-glass 


4.0x3.98 
4.0x3.98 

4 
4.5x4.55 

6 


0.024 
0.025 
0.038 
0.056 
0.059 


84 

93 

150 

280 
152 


3504 
3720 




3947 
5625 
3864 



Mean 4132 



Green-glass. 



4.95 


x5.0 


4.95 


x5.0 


4.0 


x4.05 


4.0 


x4.03 



0.022 
0.020 
0.018 
0.016 




5113 
5312 
4666 
5126 



Mean 5054 



t 2 dx 

distance is x from the centre of the sphere, is T2nrdx, — = 2r,r > 'T— ^-; and the 

total resistance is the integral of this expression between the limits of r and 
r + t. 



TENSION. 



31 




Mean 5960 



The following table exhibits the tensile strength of cylindrical 
glass bars according to the experiments of Mr. Fairbairn : — 



Description of the glass. 


Area of specimen in 
inches. 


Breaking weight in 
lbs. 


Tenacity per square 
inch. 


Annealed flint-glass. . . 
Green-glass 


(0.255 

|0.196 

0.220 

0.229 


583 
254 
639 
583 


2286 
2540 
2896 


Crown-glass 


2546 



As might have been anticipated, the tenacity of bars is much 
less than globes ; for it is difficult to make a longitudinal strain 
without causing a transverse strain, and the latter would have a 
very serious effect : it is also probable that the outer portion of 
the annealed glass is stronger than the inner, and there is a 
larger amount of surface compared with the section, in globes 
than in cylinders. 



RIVETED PLATES. 

Q7. riveted plates are used in the construction of boil- 
ers, roofs, bridges, ships, and other frames. It is desirable to 
know the best conditions for riveting, and the strength of riveted 
plates compared with the solid section of the same plates. The 
common way of riveting is to punch holes through both plates, 
into which red-hot bolts or rivets are placed, and headed down 
while hot. The process of punching strains, and hence weakens, 
the material. A better way is to bore the holes in the plates, 
and then rivet as before. The holes in the separate plates 
should be exactly opposite each other, so that there will be no 
side strain on the plates caused by driving the rivets home, and 
to secure the best effects of the rivets themselves. They are 
sometimes placed in single and sometimes in double rows, 



32 



THE RESISTANCE OF MATERIALS. 



and experiment shows that the latter possesses great advantage 
over the former. Experiments have been made upon plates of 





( 


) 


Q Q) Q 


F 

( 


© © © 







Fig. 12. 



the form shown in Fig. 12, both with lap and butt-joints, and 
with single and double rows of rivets.* 

Table showing the Strength of Single and Double Riveted Plates. 



Cohesive strength of the plates 

in lbs. per square inch. 

T. 


Strength of single-riveted joints 
of equal section to the plates, 
taken through the line of riv- 
ets. Breaking weight in lbs. 
per square inch. 


Strength of double-riveted joints 
of equal section to the plates, 
taken through the line of riv- 
ets. Breaking weight in lbs. 
per square inch. 


57,724 
61,579 
58,322 
50,983 
51,130 
49,281 


45,743 
36,606 
43,141 
43,515 
40,249 
44,715 


52,352 

48,821 
58,286 
54,594 
53,879 
53,879 


Mean.. 54,836 


42,328 


53,635 



It will be observed that in double-riveting there is but little 
loss of strength, while there is considerable loss in single-rivet- 
ing. In the preceding experiments the solid section of the 
plates, taken through the centre of the rivet-holes, was used ; 
but, as Fairbairn justly remarks, we must deduct 30 per cent. 
for metal actually punched out to receive the rivets. But as 
only a few rivets came within the limits of the experiments, 
and as an extensive combination of rivets must resist more 



* London Phil Transactions, part 2d, 1850, p. 677. 



TENSION. S3 

effectually, and as something will be gained by tlie friction 
between the plates, it seems evident that we may use more than 
60 per cent, of the strength of riveted plates as indicated above 
Fairbairn says we may use the following proportions : — 

Strength of plates 100 

Strength of double-riveted plates 70 

Strength of single -riveted plates 56 

28, STRENGTH OF DRILLED AND PUNCHED BOILER 

plates. — A committee of the Railway Master Mechanics' 
Association for 1872 reported the following results of some ex- 
periments : — 

Three pieces of t 5 q- inch boiler plate, If inch wide, were torn 
in two by hydraulic pressure. 

No. 1 broke under a strain of 32,228 lbs. 

No. 2 broke under a strain of 32,228 lbs. 

No. 3 broke under a strain of 33,600 lbs. 

The average breaking strain being. . . . 32,685 lbs. 

Three pieces of T 5 ¥ x If inch plate were punched, with a single 
f- inch hole in each piece. They were then subjected to a ten- 
sile strain, with the following result : — 

No. 1 broke under a pressure of 13,371 lbs. 

No. 2 broke under a pressure of 13,371 lbs. 

No. 3 broke under a pressure of 13,314 lbs. 

The average being 13,352 lbs. 

Three pieces of -^ x If inch plate were drilled, with a single 
•| inch hole in each piece. 

No. 1 broke under a pressure of 17,828 lbs. 

No. 2 broke under a pressure of 17,485 lbs. 

No. 3 broke under a pressure of 17,622 lbs. 

The average being 17,645 lbs. 

The average strength of the drilled plate being 4,163 lbs. 
greater than that of the punched plate. 

Great care was taken to dress the pieces to the sizes given 
after they were punched or drilled. 

The following comparative tests were then made with 
punched and drilled plates riveted. 

^ix pieces If inch wide, and cut from the same sheet as the 



34 



THE RESISTANCE OF MATERIALS. 



foregoing, were punched and riveted together in pairs with the 
best f inch rivets, one rivet to each pair, and were subjected to 
a tensile strain, with the following result : — 

No. 1 broke in centre line of hole under . . .17,828 lbs. 

No. 2 broke in centre line of hole under. . . 17,828 lbs. 

No. 3 broke in centre line of hole under.. .17,143 lbs. 
The average breaking strain being. . . . 17,599 lbs. 
Six pieces, duplicates of those last mentioned, were drilled 
and riveted together in pairs, one -f inch rivet to each pair. 

No. 1 sheared the rivet under pressure of. . 17,143 lbs. 

No. 2 sheared the rivet under pressure of. .16,457 lbs. 

No. 3 sheared the rivet under pressure of. .15,428 lbs. 
The average shearing strain being .... 16,342 lbs. 
In the last set of experiments the strength of the plates was 
not determined, since the rivets broke by shearing before the 
plates broke. It is to be regretted that the size of the rivets 
was not increased sufficiently to cause the plates to break and 
thus secure a good comparative test. It is evident, however, 
that drilled holes cause the rivets to be sheared more easily than 
punched ones. 

*&% t FAIRBASRN'S RUIiE FOR THE SIZE AND DISTRIBU- 
TION of rivets.— The best size of the rivets, the distance be- 
tween them, and the proper amount of lap of the plates, can be 
determined only by long experience, aided by experiments. 
Fairbairn gives the following table as the results of his infor- 
mation upon this important subject, to make the joint steam oi 
water tight : — 

Table shoioing the strongest Forms and best Proportions of Riveted Joints, as 
deduced from Experiments and actual Practice. (" Useful Information for 
Engineers,'''' 1st Series, p. 285.) 



Thickness of 

plates, in 

inches. 

t. 


Diameter of 

the Rivets, in 

inches. 

d. 


Length of 

rivets from the 

head, in inches. 

1. 


Distance of 
rivets from cen- 
tre to centre, in 
inches. 
a. 


Quantity of 
lap in single- 
joints, in niches. 
b. 


Quantity of 
lap in double- 
riveted joints, 
in inches, 
c. 


ft to A 

t*6- ton 


at 

u 

1* 


4*t 

tc 

u 


6t 
5t 

4t 


6t 

u 

5£t 

4it 


iOt 

u 

8it 
6gt 



TENSION. 35 

30. — STRENGTH OF IKON IN DIFFERENT DIRECTIONS 

OF THE ROLLED SHEET.* 

. In obtaining specimens for these experiments, care was gene- 
rally taken to have them cut in different directions of the roll- 
ing, longitudinally and transversely, and in some cases diag- 
onally^ to that direction. The table will be found to indicate 
the direction of slitting in each case, and the comparison 
contained in the table is given to show what information the 
inquiry has elicited. 

The comparison is made principally on the minimum strength 
of each bar, being that which can alone be relied on in practice; 
for if the strength of the weakest point in a boiler be overcome, 
it is obviously unimportant to know that other parts had greater 
strength. In one case, however, two bars, one cut across the 
direction of rolling, and the other longitudinally, were, after 
being reduced to uniform size, broken up cold, with a view to 
this question. The result showed that the length-strip was 7^- 
per cent, stronger than the one cut crosswise, considering the 
tenacity of the latter equal to 100. Of the other sets, embra- 
cing about 40 strips cut in each direction, it appears that some 
kinds of boiler iron manifest much greater inequality in the 
two directions than others. It is in certain cases not much over 
one per cent., and in others exceeds twenty, and as a mean of 
the whole series it may be stated to amount to six per cent, of 
the strength of the cross-cut bars. The number of trials on 
those cut diagonally is not perhaps sufficiently great to warrant 
a general deduction ; but, so far as they go, they certainly indi- 
cate that the strength in this direction is less than either of the 
others. 

Had we compared the mean instead of the least strength of 
bars as given in the table, the result would not have differed 
materially in regard to the relative strength in the respective 
directions. 

The boiler-iron manufactured by Messrs. E. H. & P. Ellicott, 
which was tried in all these modes of preparation of specimens, 
gave the following results : — 

1. When tried at original sections, seven experiments on 
length-sheet specimens gave a mean strength of 55285 lbs. per 

*Jour. of the Frank. Inst., Vol. 20, 2d series, p. 94. 1837. 



db THE RESISTANCE OF MATERIALS. 

square inch, the lowest being 44399 lbs., and the highest 59307 
lbs. Fourteen experiments on cross-sheet specimens gave a 
mean of 53896 lbs., the lowest result being 50212 lbs., the high- 
est 58839 lbs. ; and six experiments on strips cut diagonally 
from the sheet exhibited a strength of 53850 lbs., of which the 
lowest was 51134 lbs., and the highest 58773 lbs. 

2. When tried by riling notches on the edges of the strips, to 
remove the weakening effect of the shears, the length-sheet bars 
gave, at fourteen fractures, a mean strength of 63946 lbs., vary- 
ing between 56346 lbs. and 78000 lbs. per square inch. The 
cross-sheet specimens tried after this mode of preparation ex- 
hibited, at three trials, a mean strength of 60236 lbs., varying 
from 55222 lbs. to 65143 lbs. ; and the diagonal strips, at four 
trials, gave a mean result of 53925 lbs., the greatest difference 
being between 51428 lbs. and 56632 lbs. per square inch. 

3. Of strips reduced to uniform size by filing, four compar- 
able experiments on those cut lengthwise of the sheet gave a 
mean strength of 63947 lbs., of which the highest was 67378 
lbs., and the lowest 60594 lbs. 

From the foregoing statements it appears that by filing in 
notches and filing to uniformity, we obtain results 63946 lbs. 
and 63947 lbs. for the strength of strips cut lengthwise, differing 
from each other by only a single lb. to the square inch, and 
that by these two modes of preparation the cross-sheet speci- 
mens gave respectively 60236 lbs. and 60176 lbs., differing by 
only 60 lbs. to the square inch. This seems to prove that by 
both methods of preparing the specimens the accidental weak- 
ening effect of slitting had been removed by separating all that 
portion of the metal on which it had been exerted. Hence we 
may infer that the differences between length-sheet and cross- 
sheet specimens are really and truly ascribable to a difference of 
texture in the two directions, which will be seen to amount, in 
the case of filing in notches, to 6.15 per cent., and in that of 
filing to uniformity, to 6.26 per cent, of strength of cross -sheet 
specimens. 



TENSION. 



37 



Fable of the comparative view of the Strength of Specimens of ten different sorts of 
Boiler and one of Bar Iron, in the longitudinal, transverse, and diagonal direc- 
tion of the rolling, as deduced from the least strength of each specimen, and the. 
average minimum of each sort of Iron, in each direction in which it was tried. 



S • 


5 "3 








II 


a 


1 2 


•si 

.did 


be » . 


'So 

CD - ,J 


■q 

^••Cn-2 


4 


he? . 


D 


§«g 


a £ s 
ego 


O 


"CO 


ggg 


s^g 


6 


as 2 


+3 +a '43 


d 


|5 2 


SS-J 


£•■5% 


Z 


GQ 


CO 


£3 


go 


01 


xa 


2 


58977 




125 


57182 


Tilted. 




3 


53828 




130 


Tilted. 


57789 




4 


47167 




133 


do. 


53176 




6 




52280 


135 


do. 




47738 


.8 




50103 


137 


do. 




50358 


Mean. 


53324 


51191 


Mean. 


57182 


55882 


49048 


42 


51653 


Puddled. 


142 


44399 






43 


44102 


do. 


143 


53135 






44 


53836 


do. 


146 


60594 






46 


59262 


H'd pla.* 


148 




52468 




48 


59418 


do. 


149 




52228 




49 


57565 


do. 


150 




56869 




51 


H'd pla. 


59656 


151 




53811 




53 


H'd pla. 


56062 


152 




56073 




56 


Puddled. 


57926 


154 






51134 


58 


do. 


50570 


157 






52102 


59 


48308 


Puddled. 


160 




53862. 




60 


58648 


do. 


162 




50212 




61 


52869 


do. 


164 


56346 






62 


57612 


do. 


167 


56682 






64 


Puddled. 


45392 


169 


54361 






65 


do. 


51255 


171 






55612 


68 


57929 


H'd pla. 


174 






51425 


70 
71 


47638 
H'd pla. 


do. 
54634 










Mean. 


54253 


53646 


52568 


73 


do. 


52657 










74 


do! 


49351 


226 




49053 




Mean. 


54074 


53049 










227 
228 




53699 
40643 




















229 




46473 










230 


49368 








Mean. 


49368 


47647 








1 ' 





The specimens from 42 to 74 were partly puddled iron, and 
partly Juniata blooms, hammered and rolled into plate. The 

* Hammered and rolled into plates. 



38 THE RESISTANCE OF MATERIALS. 

length and cross-sheet specimens of these two kinds must be 
compared separately. 

All the experiments on No. 228 (cross) and 230 (length) were 
made at ordinary temperatures with a view to this comparison. 

31. — TENSIL.E STRENGTH OF AMERICAN BOILER IRON, 

as determined by Mr. F. B. Stevens at the Camden and Amboj 
K. E. repair shops, New Jersey, by sixteen experiments upon 
high grade American boiler plate, gave the following results : — 

Average breaking weight, lbs. per square inch 54,123 

Highest breaking weight, lbs. per square inch 57,012 

Lowest breaking weight, lbs. per square inch 51,813 

Variation in per centum of highest 9.1 

33. — TENSIIiE STRENGTH OF WROUGHT IRON AT VARI- 
OUS TEMPERATURES. 

Mr. Fairbairn has made experiments upon rolled plates of 
iron, and rods of rivet, iron, at various temperatures. The for- 
mer were broken in the direction of the fibre and across it. 
The specimen when subjected to experiment was surrounded 
with a vessel into which freezing mixtures were placed to pro- 
duce the lower temperatures, and oil heated by a fire under- 
neath to produce the high temperatures. The experiments 
were made upon Staffordshire plates, which are inferior to 
several other kinds in common use. The following table (A), 
gives a summary of these results. 

The mean values given in the sixth column of this Table 
exhibit a remarkable degree of uniformity in strength for all 
temperatures, from 60 degrees to 395 degrees. The single ex- 
ample at degrees gives a higher value than the mean of the 
others, but not higher than for some of the specimens at higher 
temperatures. At red heat the iron is very much weakened. 
This fact should be noticed in determining the strength of 
boiler-flues, as they are often subjected to intense heat when not 
covered with water. 

The experiments upon rivet iron were made with the same 
machine, and in the same manner, the results of which are 
shown in the following table (B). 



TENSION. 



39 



Table A — Showing Hie Resistance of Staffordshire Plates at different Tempera- 

tures. 



0) 








baa 


hug 




S 




% 




t^ 3 1 




1 

<X» 


3 


"So 


is m' 




05 M 
L, ft CO 


Remarks. 


o 

d 


1 


p 


3 


m 


§11 

« & g 




£ 


H 


m 


w 


n 


a ' 




1 


0° 


0.6868 


33,660 


49,009 


49,009 


With. 


2 


60 


0.7825 


31,980 


40,357 


) 


Across. 


3 


60 


0.6400 


27,780 


43,406 


1 44,498 


Across. 


4 


60 


0.6368 


31,980 


50,219 


i 


With. 


5 


110 


0.6633 


29,460 


44,160 


) 


Across. * 


6 


112 


0.6800 


28,620 


42,088 


1 42,291 


With. 


7 


120 


0.8128 


37,020 


40,625 


) 


With. 


8 


212 


0.8008 


31,980 


39,935 


) 


With. 


9 


212 


0.6633 


30,300 


45,680 


[ 45,005 


Across. 


10 


212 


0.6800 


33,660 


49,500 


) 


With. 


11 


270 


0.6432 


28,620 


44,020 


44,020 


With. 


12 


340 


0.6400 


31,980 


49,968 


I 46,018 


With.f 


13 


340 


0.6800 


28,620 


42,088 


Across. 


14 


395 


0.6666 


30,720 


46,086 


46,086 


With. 


15 


Scarcely red 


0.6200 


23,520 


38,032 


!■ 34,272 


Across. 
Across.^ 


1(3 


Dull red 


0.6076 


18,540 


30,512 



Table B— Showing the 



Results of Experiments on 
Temperatures. 



Bivet Iron at different 



-p* 

a 






a 
•a 
-p 


S^f 


a 3 




| 




02 


| 


■II 


% £ 




o 


J3 


El 


&| 


*g 


•°p, • 


Remarks. 


■g 

d 


p. 
S 

H 


8 

o 
o 

CO 


60 

.S 

03 

o 

P3 


bog 

Si 


a,a _ 

w II 




17 


—30° 


0.2485 


15,715 


63,239 


63,239 


Too low. 


18 


+ 60 


0.2485 


15,400 


61,971 


j- 62,816 


Too low. 


19 


60 




15,820 


63,661 


Too low. 


20 


114 




17,605 


70,845 


70,845 




21 


212 




20,545 


82,676 


) 




22 


212 


6.1963 


14,560 


74,153 


l 79,271 




23 


212 


0.2485 


20,125 


80,985 


) 




24 
25 


250 
270 


0.1963 
0.2485 


16,135 
20,650 


82,174 
83,098 


{ 82,636 




26 


310 


0.1963 


15,820 


80,570 


j- 84,046 




27 


325 


0.1963 


17,185 


87,522 




28 


415 


0.2485 


20,335 


81,830 


[ 83,943 




29 


435 




21,385 


86,056 




30 


Red heat. 




8,965 


36,076 


35,000 


Too high. 



* Too high ; fracture very uneven. 

f Too low ; tore through the eye. 

X Too high ; the specimen broke with the first strain. 



40 THE RESISTANCE OF MATERIALS. 

From this Table we see that there is a gradual increase oi 
strength from 60 degrees to 325, where it appears to attain its 
maximum. The increase is a very important amount, being 
about 30 per cent. 

It appears remarkable that the specimen at —30 degrees 
is stronger than the mean of the two at 60 degrees ; but nume- 
rous experiments which have been made by different persons 
confirm this result, when the pieces are broken by a steady 
strain. 

Experiments made by M. Baudrimont gave the following 
results : — * 

Tenacity of Iron at 32° Fahr. 205 ) ™ 

Tenacity of Iron at 212° Fahr. 191 \ S^.P 6 * S ^ are 

Tenacity of Iron at 392° Fahr. 210 J Mlllimetl ^ 

in which we observe the same general results as in the preced- 
ing Tables. 

But iron and steel will not resist shocks as effectually at very 
low temperatures as at moderate temperatures ; as we shall 
have occasion to notice more particularly hereafter. 

Mr. Johnson, when in the employ of the Navy Department, 
in 1844, made some experiments to determine the effects of 
thermo-tension upon different kinds of iron.f He took two 
bars of the same kind of iron, and of the same size, and broke 
one while cold. He then subjected the other to the same 
tension when heated 400 degrees, after which the strain was 
relieved, and the bar was allowed to cool, and the permanent 
elongation noted, after which it was broken by an additional 
load. It will thus be seen that the experiments were not con- 
ducted in the same way as those by Fairbairn. Table A, page 
41, gives the results of his experiments. 

Remarks. — From the two former sets of experiments, p. 
39, it appears that the strength of the iron was increased 
by an increase of temperature at the time the bar was 
broken, and by the latter that it was not only increased, but> 
by being subjected to severe tension while at a high temper- 
ature, the increased strength was not lost by cooling. It hardly 
seems probable that this increased strength would be retained 

* Jour. Frank Inst., Vol. 20, 3d series, p. 344. 

f Senate Doc, No. 1, 28th Cong., 2d Sess., 1844-5, p. 639. 



TENSION. 



41 



indefinitely, and hence it would be important to know how long 

it was after the piece was cooled before it was broken. 

Table A — Results of Experiments on Thermo- Tension, at 400° Temperature. 



KIND OP IKON. 


2 


u 

""A . 
£ 2 




ength lin- 
mo-tension 
train equal 
strength 
Id. 


i 


11 

* a 

e!'<a 




J3 

a 
w ■ 




a 
.2 
o 
<v 
m 


Gain of 1 
der ther 
with a s 
to the 
when co 


a 
"3 


Total gai 
to the 
section. 




Tons. 


Tons. 


Inches. 


Per cent. 


Per cent. 


Per cent. 


Tredegar, round . . . 


60 


71.4 


1.91 


6.51 


19.00 


25.51 


Tredegar, round . . . 


60 


72.0 


1.91 


(6.51) 


20.00 


26.51 


Tredegar, square bar 


60 


67.2 


1.69 


6.77 


12.00 


18.77 


Tredegar, r'nd, No. 3 


58 


68.4 


1.15 


5.263 


17.93 


23.19 


Salisbury, round . . . 


105.87 


121.0 


3.59 


3.73 


14.64 


18.37 


Me 


an 






5.75 


16.64 


22.40 



These results are confirmed by the experiments of the com- 
mittee of the Franklin Institute, as shown by the following 
Table. — See Journal of the Franklin Institute, Yol. 20, 3d 



ISeries, p. 22. 



ABSTRACT OF TABLE 



Of the comparative view of the Influence of High Temperatures on the strength of 
Iron, as exhibited by 73 experiments on 47 different specimens of that metal at 
46 different temperatuns, from 212 3 to 1317° Fahr., compared with the 
strength of each bar when tried at ordinary temperatures, the number of 
experiments at the latter being 163. 



No. of the experi- 


Temperature observed 


Strength at ordinary 


Strength at the tem- 


ment. 


at moment of fracture. 


temperature. 


perature observed. 


1 


212° 


56736 


67939 


2 


214 


53176 


61161 


3 


394 


68356 


71896 


9 


440 


49782 


59085 


10 


520 


54934 


58451 


15 


554 


54372 


61680 


20 


568 


67211 


76763 


25 


574 


76071 


65387 


40 


722 


57133 


54441 


45 


824 


59219 


55802 


50 


1037 


58992 


37764 


58 


1245 


54758 


20703 


59 


1317 


54758 


18913 



42 THE RESISTANCE OF MATERIALS. 

Remark. — According to these experiments, as shown in the 
fourth column, the strength increases with the temperature to 
394 degrees, when it attains its maximum ; although in some 
cases the strength was increased by increasing the temperature 
to 568 degrees. By comparing the third and fourth columns 
we see that the strength is greater for all degrees from 212° to 
574° than it is at ordinary temperatures, but above 574° it 
is weaker. The experiments on Salisbury iron showed that the 
maximum tenacity was 15.17 per cent, greater than their mean 
strength when tried cold. The committee above referred to 
determined the maximum strength of about half the specimens 
used in the preceding Table by actual experiment, and calculated 
it for the others ; and from the results derived the following 
empirical formula for the diminution in strength below the 
maximum for high degrees of heat : — 

jD* = o(d - 80) 13 
in which D is the diminution after it has passed the maximum, 
6 the temperature Fahrenheit, and 
c a constant. 

The value of the constant in empirical formulas is not strictly a 
constant, but is the mean of several values which are considered 
as constant. The value of the constant is found by substitut- 
ing known values for all the other quantities in the equation. 

This formula appears to be sufficiently exact for all tempera- 
tures between 520° and 1317°. 

33. "^ENSIUE STRENGTH OF OTHER METALS AT DIF- 
FERENT temperatures.— Experiments made by M. Bau- 
drimont* showed that for all the following named metals the 
strength diminished as the temperature was increased; the 
results of which are given in Table A, page 43. 

34. EFFECT OF SEVERE STRAINS UPON THE ULTIMATE 

tenacity of iron Rous. — Thomas Loyd, Esq., of England, 
took 20 pieces of If S.C. <^j|> bar iron, each 10 feet long, which 
were cut from the middle of as many rods. Each piece was cut 
into two parts of 5 feet each, and marked with the same letter. 
Those marked A, B, 0, &c, were first broken, so as to get the 
average breaking strain. Those marked A2, B2, &c, were 

* Jour. Frank. Inst., Vol. 20, 3d Series, p. 344, 1850. 



TENSION. 



43 



subjected to the constant action of three-fourths the breaking 
weight, previously found, for five minutes. The load was then 
removed, and the rods afterwards broken. The results are 
given in Table (B). 

Table (A) — Of the Mean Values of the Tenacity of the principal Malleable Metals 
at the temperature #/ 32°, 212° and 392° Fahrenheit. 



Name of the Metal. 


Tenacity per square 
Millim. per cross section. 




0° 


212° 


392° 


Gold 


18.400 
22.625 
25.100 
28.324 
36.481 


15.224 

19.284 
21.873 
23.266 
32.484 




12.878 


Platina 


17.277 


Copper 


18.215 


Silver 


18.577 


Palladium. 


27.077 







Table B. — Results of the Experiments.* 



FIRST. 


SECOND. 


Mark on the bars. 


Breaking weight in 
tons (gross). 


Mark. 


Breaking weight in 
tons. 


A 
B 

D 
E 
F 

a 

H 
I 
J 
K 
L 
M 
N 
O 
P 
Q 
R 
S 
T 


33.75 
30.00 
33.25 
32.75 
32.50 
33.25 
32.75 
33.25 
33.50 
33.50 
32 25 
32.25 
30.25 
34.25 
31.75 
29.75 
33.50 
33.75 
33.00 
32.25 


A 2 
B 2 
C 2 
D 2 
E 2 
F 2 
G 2 
H 2 
I 2 
J 2 
K 2 
L 2 
M 2 
N 2 
O 2 
P 2 
Q 2 
R 2 
S 2 
T 2 


33.75 
33.00 
33.25 
32.25 
32.50 
33.00 
33.00 
33.50 
32.75 
33.25 
32.50 
31.50 
32.75 
34.00 
32.50 
31.00 
33.75 
33.75 
33.25 
31.00 


Mean 


32.57 




32 81 







Fairbairn, Useful Information for Engineers, First Series, p. 313. 



44 



THE RESISTANCE OF MATERIALS. 



We here see that a strain of 25 tons, or three-fourths the 
breaking weight, did not weaken the bar. 

These experiments indicate that a frame maybe subjected to 
a severe strain of three-fourths of its strength for a very short 
time without endangering its ultimate strength. 

3,5. EFFECT OF REPEATED RUPTURE. The following 

experiments were made at Woolwich Dockyard, England. The 
same bar was subjected to three or four successive ruptures by 
ten- ile strains. They show the remarkable fact that while great 
strains impair the elasticity, as shown by Hodgkinson, yet they 
do not appear to diminish the ultimate tenacity. 



Table shotting the effect of repeated Fracture on Iron Bars. 





First breakage. 


Second breakage. 


Third breakage. 


Fourth breakage. 


Eeduced 
from 






















Mark. 




Stretch 




Stretch 








Stretch 


area of 
1.37 sqr. 




Tons. 


in 54 


Tons. 


hi 36 


Tons. 


in 24 


Tons. 


in 15 


inches to 






inches. 




inches. 




inches. 




inches. 


the fol- 
lowing. 






In. 




In. 




In. 








A 


83.75 


0.9125 


35.50 


0.200 












B 


33.75 


0.9250 


35.25 


0.225 


37.00 


1.00 


38.75 




1.25 


B 


32.50 


0.9250 


34.75 


0.125 












P 


33.25 


1.0500 


35.50 


0.112 


37.25 


0.62 


40.40 




1.18 


G- 


32.75 


0.8500 


35.00 


0.125 


37.50 




40.41 




1.25 


H 


33.75 


1.0625 


36.25 


0.187 












I 


33.50 


0.S375 


34.50 


0.62 


36.50 


1.50 








J 


33.50 


0.9250 


36.00 


0.025 


36.75 


1.12 


41.75 




1.25 


L 


32.25 


Defect'e 


36.50 


0.150 


37.75 




41.00 


0.31 


1.25 


M 


30.25 


Defect' e 


36.50 


.62 


37.75 


0.60 


38.50 


0.06 


1.25 




32.95 




35.57 




37.21 




40.16 




1.24 






Mean per s \. in . 


24.04 




25.93 




27.06 




29.20 




0.90 



We thus see that while the section is reduced 10 per cent., the 
strength is apparently increased over 20 per cent. It is not, 
however, safe to infer that the strength is actually increased, 
for it is probable that it broke the first time at the weakest 
point, and the next time at the next weakest point, and so on. 

We also observe that the total elongations are not propor- 
tional to the tensile strains, which is in accordance with the 
results of other experiments. 



TENSION. 



45 



ANNEALED METAL STRENGTH OF. 

36. annealing is a process of treating metals so as to make 
them more ductile. To secure this, the metals are subjected to 
a high heat and then allowed to cool slowly. Steel is softened 
in this way, so that it may be more easily worked. Campin* 
says that steel should not be overheated for this purpose. Some 
bury the heated steel in lime ; some in cast-iron borings ; and 
some in saw-dust. He (Campin) says the best plan is to put 
the steel into an iron box made for the purpose, and fill it with 
dust-charcoal, and plug the ends up to keep the air from the 
steel ; then put the box and its contents into a fire until it is 
heated thoroughly through, and the steel to a low red heat. It 
is then removed from the fire, and the steel left in the box until 
it is cold. Tools made of annealed steel will, in some cases, 
last much longer than those made of unannealed steel. 

But it appears from the following Table that it weakens iron 
to anneal it. 



Table of 


the Strength of Wrought Iron Annealed at Different Temperatures^ 




Strength at or- 




Strength at 


Strength after 


Ratio of di 


No. of com- 


dinary temper- 


Temperature at which 


the annealing 


annealing and 


minution i/r 


parisons. 


ature before an- 
nealing. 


annealing took place. 


temperature. 


cooling. 


strength. 


1 


57,133 


1037° 


37,764 


55,678 


0.025 


5 


53,774 


1155 


21,967 


45,597 


0.152 


10 


52,040 


1245 


20,703 


38,843 


.253 


15 


48,407 


Bright welding- heat. 




38,676 


.201 


17 


73,880 


Low welding heat. 




54,578 


.275 


18 


76,986 


Bright welding heat. 




50,074 


.349 


19 


89,162 


Low welding heat. 




48,144 


.460 



37. THE STRENGTH OF IRON AND STEEL, ALSO DEPENDS 

largely upon the process of their manufacture and their treat 
ment afterwards. The strength of wrought iron depends upon 
the ore of which it is made ; the manner in which it is smelted 
and puddled ; the temperature at which it is hammered, and the 
amount of hammering which it receives in 



bringing it into 



* Campin's Mechanical Engineering. 

t Jour. Frank. Inst., Vol. 20, 2d Series, p. 109, 1837. 



46 THE RESISTANCE OF MATERIALS. 

shape. The same remark applies to cast-steel. If the formei 
is hammered when it is comparatively cold, it will weaken it, 
especially if the blows are heavy ; but the latter, steel, may be 
greatly damaged, or even rendered worthless by excessive heat, 
and it is greatly improved by hammering when comparatively 
cold. For the effect of tempering on the crushing strength, see 
Article 59. 

Different ores with essentially the same treatment produce 
essentially different iron. Thus, the Lake Superior ores, near 
Marquette, make a soft but very tough iron. Some of the 
strongest specimens of iron which have been made in this 
country were made from these ores, but it is found that the 
elastic limit is passed with a much less strain compared with its 
ultimate strength than many other irons. Manufacturers, 
therefore, mix it with other ores so as to raise the elastic limit. 
They often mix it with cheaper ores so as to cheapen the pro- 
duct. They also mix it with cheaper ores so as to improve the 
quality of the iron which would result if cheap ores only were 
used. The mixing of ores from various mines is constantly 
going on among manufacturers for various reasons. In this 
way is secured irons of various grades of hardness, of elasticity, 
of weldibility, and of tenacity. 

There is even a greater difference in the quality of steel than 
of wr ought-iron. We have the well-known classes of blister 
steel, crucible steel, Bessemer steel, and more recently of 
chrome steel. Uniformity of product is more earnestly sought 
in the manufacture of the several grades of steel than of iron, 
but when the same iron is used by the same person, under the 
same conditions, so far as he is able to control them, the expert 
finds that there is a perceptible difference in the products. 
Some steel takes a higher temper than others; some is softer; 
some more brittle; some more tenacious; some is better to resist 
crushing ; some better for sharp tools ; some better to work into 
masses than others; and it is often necessary for those who use 
steel to become acquainted with the grade which will best suit 
their purpose. 

Although experts may detect differences in steels of the same 
general grade, yet manufacturers are able to produce stee] 
having given general characteristics so uniform that the com 
mon workman will not detect any difference in them. 



TENSION. 47 

,18. chkome STEEii. This is a peculiar product, which, 
according to the older definitions of steel (depending upon a 
certain per cent, of carbon), is not steel, but which possesses 
many of the characteristics of steel. The manufacturers of it 
claim that they can produce a steel of more uniform quality 
of any particular grade, especially in large masses, than can be 
produced by carbon steel. 

The tensile strength exceeds considerably that of the best 
crucible steel (excepting the remarkable specimen noted on p. 
25). The experiments which were made upon twelve speci- 
mens of tool steel, which were cut from three bars, at the West 
Point Foundry, gave the following results : — 

Highest strength 198,910 lbs. per square inch. 

Lowest strength 163,760 lbs. per square inch. 

Average of all 179,980 lbs. per square inch. 

The limit of elastic resistance is also high, being more than 
half of its ultimate strength. * 

39. prolonged fusion of cast iron. — Cast iron is 
also subjected to great modifications of strength on account of 
the manipulations to which it is or may be subjected in its 
manufacture and preparations for use. The strength in some 
cases is greatly increased by keeping the metal in a fused state 
some time before it is cast. Major Wade made experiments 
upon several kinds of iron, all of which were increased in 
strength with prolonged fusion (see Rev., p. 44), one example 
of which is given in the following 

Table allowing the Effecta of Prolonged Fuaion. 





Tensile Strength in 
lbs. per sq. in. 


Iron in fusion 


4 hour 


17,843 


Iron in fusion 


1 hour 


20,127 
24,387 
34,496 


Iron in fusion 




Iron in fusion 


2 hours 





4@. effect of REMELTisG cast iron. — But the great- 
est effect was produced by remelting. The density, tenacity, 

* Report of Capt. J. B. Eads, C.E. This steel is used in the construction of 
the noted St. Louis bridge. 



48 



THE RESISTANCE OF MATERIALS. 



and transverse strength were all increased by it, within certain 
limits. For instance, a specimen of No. 1 Greenwood pig- ire n 
gave the following results. (12ep., p. 279.) 



Table shotting the Effects of Bemelting. 




No. 1 Greenwood Iron. 


Specific 

gravity. 


Tensile Strength. 


Crude pig-iron 


7,032 

7,086 
7,198 
7.301 


14,000 


Crude remelted once 


22,900 


Crude remelted twice 


30,229 


Crude remelted three times 


35,786 





But there is a point beyond which remeltings will weaken 
the iron. Mr. Fairbairn made an experiment in which the 
strength of the iron was increased for twelve remeltings, and 
then the strength decreased to the eighteenth, where the experi- 
ment terminated. In some cases no improvement is made by 
remelting, but the iron is really weakened by the process ; so 
that it becomes necessary to determine the character of each 
iron under the various conditions by actual experiment. 

The laws which govern Greenwood iron were so thoroughly 
determined that the results which will follow from, any given 
course of treatment may be predicted with much certainty 
(Eep., p. 245). 

By mixing grades Nos. 1, 2, and 3, and subjecting them to a 
third fusion, one specimen was obtained whose density was 
7,304, and whose tenacity was 45,970 lbs., which is the strong- 
est specimen of cast iron ever tested. {Rep., p. 279.) 

As a general result of these experiments, Major Wade re- 
marks (p. 243), " that the softest kinds of iron will endure a 
greater number of meltings with advantage than the higher 
grades. It appears that when iron is in its best condition for 
casting into proof bars (that is, small bars for testing the metal) 
of small bulk, it is then in a state which requires an additional 
fusion to bring it up to its best condition for casting into the 
massive bulk of cannon." 

41. the manner of cooiiiNG also affects the strength. 
It was found that the tensile strength of large masses was 
increased by slow cooling; while that of small pieces was 
increased by rapid cooling. (JSep., p. 45.) 



TENSION. 49 

42. THE JHODULVS OF STRENGTH IS MODIFIED, we th.US 

Bee, by a great variety of circumstances ; and hence it is im- 
possible to assign any arbitrary value to it for any material 
that will be both safe and economical; but its value must be 
determined, in any particular case, by direct experiment, or 
something in regard to the quality of the material must be 
known before its approximate value can be assumed. 

43. safe limit of loading. — Structures should not be 
strained so severely as to damage their elasticity. According 
to Article 9, it appears that a weight suddenly applied will 
produce twice the elongation that it will if applied gradually 
or by increments. Hence, structures which are subjected to 
shocks by sudden applications of the load, should be so propor- 
tioned as to resist more than double the load as a constant, 
dead-weight without straining it beyond the elastic limit. 

This method of indicating the limits is. perfectly rational; 
but, unfortunately, the elastic limits have not been as closely 
observed and as thoroughly determined by experimenters as 
the limit of rupture. The latter was formerly considered 
more important, and hence furnished the basis for determining 
the safe limit of the load. Observations on good constructions 
have led engineers to adopt the following values as mean re- 
sults for permanent strains in bars : — 

I™ r^t ^ i di I ° f tne ioad wnicn woiiid 

For wrought iron, \to\\ ^ runture 
For cast iron, \ to \ ) P rocmce rupture. 

Further observations will be made upon this subject in the 
latter part of the volume. 

4r 



60 



THE RESISTANCE OF MATERIALS. 



CHAPTER II. 

COMPKESSIOE". 



ELASTIC RESISTANCE. 

44. — compression of cast iron. — Cap tain T. J.Eodman, 
in his Eeport upon metals for cannon, page 163, has given the 
results of experiments upon a piece of cast iron, which was 
taken from the body of the same gun as was the specimen re- 
ferred to on page 11 of this work, the results of which are given 
in the following Table. 

TABLE 

Showing the Compression, permanent Set, and coefficient of Elasticity * of a 
10 inches long and 1.382 inch diameter. 



Weight per 
square inch of 
section in lbs. 


Compression per 


Permanent set per 


Coefficient of 


inch of length. 


inch of length. 


elasticity. 


1,000 


0.000090 


0. 


11,111,000 


2,000 


0.000170 


0. 


11,824,000 


3,000 


0.000255 


0.000005 


11,843,100 


4,000 


0.000320 


0.000015 


12,500,000 


5,000 


0.000385 


0.000025 


12,987,000 


6,000 


0.000455 


0.000030 


13,189,000 


7,000 


0.000505 


0.000035 


13,861,300 


8,000 


0.000575 


0.000045 


13,813,000 


9,000 


0.000645 


0.000055 


13,952,000 


10,000 


0.000705 


0.000070 


14,196,000 


15,000 


0.001035 


0.000170 


14,492,000 


S0,00C 


0.001395 


0.000300 


14,337,000 


25,000 


0.001825 


0.000495 


13,687,900 


30,000 


0.002380 


0.000820 


12,602,300 



"We observe that the coefficient of elasticity is much less for 
the first strains than for those that follow. It thus appears that 
this metal resists more strenuously after it has been somewhat 
compressed than at first. The coefficient of elasticity is con- 

* The author computed the coefficients of elasticity from the other data of 
the table. 



COMPRESSION. 



51 



3iderably less than for the corresponding piece, as given on page 
11. The difference is very much greater than that found by 
Mr, Hodgkinson in the specimens which he nsed in his experi- 
ments. He took bars 10 feet long, and about an inch square, 
and fitted them nicely in a groove so that they could not bend, 
and occasionally, during the experiment, they were slightly 
tapped to avoid adherence. The metal was the same kind as 
that used in the experiment recorded on page 13. 



TABLE 

Giving the BesulU of Experiments ly Mr. Hodgkinson on bars of Cast Iron ten 

feet long. 



Pressure per 


Compression pei 


inch of length. 


Coefficient of 


Error in parts of 


square inch of 






elasticity per 
square inch. 


P of the f ormtila 


section. 
P. 


Total. 


Permanent. 


P=170,763(t c 
-36,318/1* 


lbs. 


in. 


in. 


lbs. 




2064.74 


0.0001561 


0.00000391 


13,231,300 


- 3V 


4129.49 


0.0003240 


0.00001882 


12,764,910 


— 8 ig" 


6194.24 


0.0004981 


0.00003331 


12,442,300 


-4- -J- 
+ 62 


8258.98 


0.0006565 


0.00005371 


12,585,100 


+ T3W 


10323.73 


0.00082866 


0.00007053 


12,467,100 


+ !\* 


12388.48 


0.00100250 


0.00009053 


12,357,200 


+ 2^8 


14453.22 


0.00128025 


0.00011700 


12,253,700 


+ lh 


16517.97 


0.00136150 


0.00014258 


12,141 ;200 


+ T k 


18582.71 


0.00154218 


0.00017085 


12,058,100 


+ lh 


20647.46 


0.00171866 


0.00020685 


12,021,800 


+ frhr 


24776.95 


0.00208016 


0.000.36810 


11,920,000 


i_ 

176 


28906.45 


0.00247491 


0.00045815 


11,687,400 


-d-5 


33030.80 


0.0029450 


0.00050768 


11,222,750 


+ 64 


37159.65 


0.003429 









In this case the highest coefficient of elasticity results from 
the smallest strain which is recorded. The difference in this 
respect between this example and the preceding one results 
doubtless from the internal structure of the iron. The coeffi- 
cient in both these cases is much less than that found for other 
kinds of cast iron, as is shown in the Table of Resistances in the 
Appendix. 

Mr. Hodgkinson proposed the empirical .formula, P =- 
170,763\ c — 36,318\c, to represent the results of the experi- 
ments ; and although it may represent more nearly the results 
for a greater range of strains than equation (3), yet there is no 
advantage in its use in practice. 



52 



THE RESISTANCE OF MATERIALS. 



45. COMPRESSION OF WROUGHT IRON. 

Mr. Hodgkinson also made experiments upon bars of wrought 
iron in precisely the same manner as upon those of cast iron, 
the results of which are given in the following 

TABLE, 

Owing the Results of Experiments by Mr. E. Hodgkinson on bars of Wrought Iron, 
each of which was ten feet long. * 





1st Bar. 


2d Bar. 


Weight producing 


Section = 1.025x1.025 sq. in. 


Section = 1.016x1.02 sq.in. 


the compression. 












Amount of 




Amount of 






Compression. 


Value of E. 


Compression. 


Value of E. 


* 
lbs. 


inch. 


lbs. 


inch. 


lbs. 


5098 


0.028 


20,796,500 


0.027 


21,864,000 


9578 


0.052 


21,049,000 


0.047 


23,595,000 


14058 


0.073 


21,979,000 


0.067 


24,273,000 


16298 


0.085 


21,343,000 






18538 


0.096 


22,156,000 


0.089 


24,108,000 


20778 


0.107 


22,160.000 


0.100 


24,038,000 


23018 


0.119 


23,587,000 


0.113 


23,587.000 


25258 


0.130 


22,095,000 


0.128 


23,679,000 


27498 


0.142 


22,111,000 


0.143 


22,259,000 


29738 


0.152 


21,938,000 


0.163 


21,139,000 


31978 


0.174 


20,979,000 


0.190 


19,478,000 


In -£• hour. 






0.261 




Again after i 










hour. 






0.269 




Then repeated. 






0.328 





4®. graphical representation. — These two cases are 
graphically represented in Fig. 13. It is seen from the tables 
that the compressions are quite uniform for a large range of 
strains, and hence equation (2), page 17, is applicable to com- 
pressive strains when within the elastic limits. In the case of 
the wrought-iron bars, the first one attains its maximum coeffi- 
cient of elasticity for a strain somewhat less than one-half its 
ultimate resistance to crushing, and the second bar at about one- 
third its ultimate resistance. 



47. COMPARATIVE RESISTANCE OF CAST AND WROUGHT 

iron. — The coefficient of elasticity is a measure of the com- 
pressibility of metals. Hence, an examination of the two pre- 
ceding Tables shows that of the specimens used in these 

* The coefficients of elasticity were computed by the author. 



COMPRESSION. 



53 























/ 






30000 






































7/7 






















// 


V 
























/ 
























/ 














30000 












/ 






















/ 


























// 
























// 


























// 


















10000 






A 






















/ 






































































lis. 

2000 



















































002 in. 



0-10 

Fig. 13. 



■20 



experiments, the cast iron was compressed nearly twice as much 
as the wrought iron for the same strains. An examination of 
the Table of Resistances, in the Appendix, shows that for a mean 
value wrought iron is compressed about two-thirds as much as 
cast iron for the same strain. The same ratio evidently holds 
for tension. This is contrary to the popular notion that cast 
iron is stiver than wrought iron ; for it follows from the above 
that a cast-iron bar may be stretched more, compressed more, 
and bent more, than an equal wrought iron one with the same 
force nnder the same circumstance, and in some cases the 
changes will be twice as great. One reason why cast is con- 
sidered stiffer than wrought iron probably is, that wrought iron 
does not fail suddenly as a general thing, but it can be seen tc 
bend for a long time after it begins to break ; while cast iron 
on account of its granular structure, fails suddenly after it be 
gins, and the bending which has previously taken place is not 
noticed. It is not safe to trust to such general observations for 



54 THE KESISTANCE OF MATERIALS. 

scientific or even practical purposes, but careful observations 
must be made, so that all the circumstances of the vwe may be 
definitely known. It will hereafter be shown that the ultimate 
resistance to crushing of cast iron is double that of wrought 
iron, and yet Fairbairn and other English engineers have justly 
insisted upon the use of wrought iron for tubular and other 
bridges For, without considering the comparatively treacherous 
character of cast iron when heavily loaded, it appears that 
within the elastic limits (and the structure should not be loaded 
to exceed that), a wrought iron structure is stiffer than a cast 
iron one of the same dimensions, and will sustain more within 
the elastic limits for a given compression, extension, or deflec- 
tion. 

48. compression of stee^. — Good cast steel has a higher 
coefficient of elasticity than any other metal upon which experi- 
ments have been made for the purpose of determining it ; and 
yet it exceeds by only a small amount the coefficient for the best 
iron. But the limit of elasticity of steel greatly exceeds that 
of iron, as has already been observed in article 38. In the noted 
St. Louis Bridge the coefficient of elasticity of the steel was not 
to be less than 26,000,000 lbs., nor exceed 30,000,000 lbs. 

49. compression of other materials. — All materials 
are compressible as well as extensible, and it is generally as- 
sumed that their resistance to compression, within the elastic 
limits, is the same as for extension ; but, as has been seen in 
the previous articles, this is not rigorously correct. The mean 
value, however, of the coefficient of elasticity is sufficiently 
exact for practical cases. 



-Example. — 1 . Kequired the compression of a sphere which rests up- 
on a plane ; the weight of the sphere being the only load. 
This may be readily solved by supposing that all horizon- 
tal sections before compression remain plane and horizon- 
tal during compression. 

Take o, the highest point of the sphere, as the origin of 
Fig. 14. co-ordinates ; x vertical and y horizontal. 

6 — the weight of a unit of volume ; and 
r = the radius of the sphere. 
Then y' 1 = 2 r %—x 2 ; 




COMPRESSION. 55 

JT=iry , = ff(2raj — x 3 ) = the area of any horizontal section ; and 

r r* 



6 J Kdx = S* J @rx 



the weight of all the segment above the section. 
/*# nx 

I Pdx_± I **f-**t *. 
• x = / E~K~ JS I »(8r»- 
U o %J o 

/x 
r 



^rf-J 


■KX* 


w (2 r x — 


•4) 


S*x 




x* dx 
2 r — x 





x dx 8 



* / . ~ . - ■ . - ; 2 r 



^1$ ( « 2 ~ 2 r a; + 4 r 2 Nap. log, 



2 r—Xj 



For a hemisphere this becomes, by making x = r; 

\ = -£4 ( 4 -y«P- %• 2 - 1) = 0.29543 ^ 



6 m K ^ *' *' ' e 

For the sphere x = 2 r and the Equation gives 

X = oo 

as it should, since theoretically there is only a point where it touches the plane 
to sustain the whole sphere, and Eq. (2) gives A = oo when k = 0, and the other 
quantities are finite. But practically we know that this is not true, and it is 
easily accounted for by supposing that the sphere as well as the body upon 
which it rests is flattened in the vicinity of x so as to present a surface of finite 
magnitude for supporting the weight above it. 

51. Required the compression of any portion of a cylindrical annulus when 
it lies upon a horizontal plane, its axis being parallel to the plane, and the 
weight of the annulus being the only load. 

The true distortion in this case, as well as in the preceding one, is peculiar. 
There will be a bulging outward, as well as depression vertically, and there 
will also be a moment of stress. But it may be solved by assuming that the 
only strains are vertical compressions, and that horizontal sections remain 
horizontal during compression. 

If the annulus is very thin compared with the diameter of the circle we 
have, the origin being at the highest point, x vertical and y horizontal; 

t = the thickness ; 

- '» 

i J =^ ( Jrverssm — - : 
r 

I = dx = the height of an infinitely short prism ; and 

T 

K = t = the horizontal section at any point for a unit of 

length. 



56 THE RESISTANCE OF MATERIALS. 



•""""*/ 



yj 2 r x — x* ve r8 sin ?Ldx 
o r 

The approximate value of which may be found by developing it into a series 
and integrating several terms. If the origin of co-ordinates be taken at the 
centre of the cylinder we have 



r 



. - X 

y r* — x* cos —dz 



From this example we see that if a large cylinder (as for instance a steam 
sylinder, or a boiler) be made exactly cylindrical when it stands upon one end, 
it will be oval when it is placed on its side. 

52. THE PARAL.L.EL.ISM: OF SECTIONS, which was assumed in the 
two preceding problems, would not be realized in any actual case. The solution 
properly belongs to the Mathematical Theory of Elasticity, and involves the 
most refined analysis. An exact solution may not be possible. 

53. A GENERAL STATEMENT OF THE PROBLEM of the math- 
ematical theory of the equilibrium of a solid body is : 

A solid of any shape, when undisturbed, is acted on in its substance by a force 
distributed through it in any manner, and displacements are arbitrarily pro- 
duced. It is required to find the displacement of every point of its surface. 

54. ANALYTICAL EXPRESSION. If X, T, and Z be the resolved 
components of the applied force, and. the remaining notation be as given on 
page 217, then for any point (x, y, z) within the solid we have 

dp** [ dp xy dp X z [ x _ Q 
dx dy dz 

dpj^ d^jy dp yz Y __ ^ 
dx dy dz 

dp iX dp ZJ dp zz 
— — + — — + -— — + Zi = u 
dx dy dz 



55. PARTICULAR VALUES. It will be shown in Chapter IX. that 
when a prismatic bar is compressed by a longitudinal stress it will expand 
laterally, and in a perfectly homogeneous body the expansion per unit will be 
approximately £ of the contraction per unit. 



COMPRESSION. 57 



ULTIMATE STBENGTH. 

5©. modulus for chmjshing. — The modulus of resistance 
to crushing is the pressure which is necessary to crush a piece 
of any material whose section. is unity, and whose length does 
not exceed from one to five times its diameter. 

The law of resistance to crushing is not simple. Granular 
blocks, like some kinds of stone and cast iron, often separate in 
planes (or surfaces approximating to planes) which are inclined 
to the base. 

Glass in some cases separates in thread-like filaments when it 
is crushed. Wrought iron does not fail suddenly, like the 
bodies just mentioned, but considerable tenacity remains be- 
tween the fibres after it begins to fail. Then, too, in all cases 
the resistance to crushing depends upon the length of the piece. 
If the blocks are very short (from one to five times the diameter 
as mentioned above) we get simply crushing ; but if they are 
long compared with their diameter, the phenomena are very 
complex, there being a combination of bending and crushing, 
and the law which governs it is determined only approximately 
by direct experiment, as indicated in article 62. 

It is found by experiment that the resistance of short pieces 
(blocks) to crushing varies nearly as the transverse section of 
the piece, no matter what the form of the fracture may be. 

Hence, if 

P — the crushing force, and 

K = the section under pressure, we have 

P= OK (22) 

modulus of strain. — If the force P is not sufficient to 
crush the piece, we have for the strain on a unit of section 

<?, = X (23) 



58 



THE BESISTANCE OF MATERIALS. 



57. — RESISTANCE TO CRUSHING OF CAST-IRON. 

TABLE 

Of the results of Experiments on the Tensile and Crushing Resistances of Cast' 
Iron of various kinds, made by Eaton Hodgkinson. * 



Description of the Iron. 



Low Moor Iron, No. 1 . 
" " No. 2. 

Clyde Iron, No. 1 

" " No. 2 

" " No. 3 

Blaenavon Iron, No. 1 . 

" No. 2. 

" " No. 3. 

Calder Iron, No. 1. 

Coltness Iron, No. 3.. , 

Brymbo Iron, No. 1 . . . 

" " No. 3... 

Bowling-, No. 2 



Ystalyfea, No. 2. 
(Anthracite) 



Yniscedwyn, No. 1 . . . 
(Anthracite) 



" No. 2.... 

Stirling, 2d quality. . . 

a 3d quality. . . 



Mean 



Tensile 

Strength per 

square inch. 

T. 



Lbs. 

12,694 
15,458 
16,125 
17,807 
23,468 
13,938 
16,724 
14,291 
13,735 
15,278 
14,426 
15,508 
13,511 
14,511 

13,952 

13,348 
25,764 
23,461 



16,303 



Height of 
Specimen. 



inch, 
t 

I 

f 

1+ 
* 

f 
l* 

I 
H 

ft 
H 

* 

4 

f 
1* 

f 
H 

ft 
H 

ft 

n 

I 

n 

t 

4 
ft 



{:: 



Crushing 

strength per 

square inch. 

C. 



Lbs. 

64,534 

56,445 
99,525 
92,332 
92,869 

88,741 

109.992 

102,030 

107,197 

104,881 

90,860 

80,561 

117,605 

102,408 

68,559 

68,532 

72,193 

75,983 

100,180 

101,831 

74,815 

75,678 

76,133 

76,958 

76,132 

7.3,984 

99,926 

95,559 

83,509 



77,124 
75,369 
125,333 
119,457 
158,653 
129,876 



88,800 
94,730 



Ratio of Tenacity 

to crushing. 

C-r-T. 



Mean. 



5.084) 1 
4.446 C 1 
6.438 
5.973 
5.759) 1 
5.503 J" 1 
6.177 J 1 
5.729 f 1 
4.568) i 
4.469 f 1 
6.519) , 
5.780 f 1 
7.032) ., 
6.123 f 1 
4.797) ., 
4.795 f 1 
5.256 
5.532 
6.557 ) 1 
6.665 y 1 
5.186} 1 
5.264 f 1 
4.909 
4.963 
5.635 
5.476 



6.585 



4.765 
6.205 
5.631 
5.953 
4.518 
6.149 
6.577 
4.796 
5.394 
6.611 
5.216 
4.936 
5.555 
6.735 



1 : 5.985 
1 : 5.638 



1 : 5.811 



5.778) . 
5.646 f 1 
4.865 ) 1 
4.637 J" 1 
6.762 f 1 
5.536 h 



5.712 
4.751 
6.149 



Meaniatio 1 : 5.64 



* Supplemmi to Bridges, by Geo. E. Brunell, and Wm. T. Clark. John 
Weale, London. 



COMPKESSION. 



59 



In this table the ratio of resistances range from about 4r| 
(Clyde, No. 3) to more than 7 (Blaenavon, No. 2). The same 
experimenter once obtained the ratio of 8.493 from a specimen 
of Carron iron, No. 2, hot blast ; * and the mean of several ex- 
periments, made at the same time, gave 6.594. Hence we have, 
as the mean result of a large number of experiments, that the 
crushing resistance of cast iron is about 6 times as great as its 
tenacity ; but the extremes are from 4J- to 8 -J times its tenacity. 

58. RESISTANCE OF WROUGHT IRON TO CRUSHING. — 

Comparatively few experiments have been made to determine 
how much wrought iron will sustain at the point of crushing, 
and those that have been made give as great a range of results 
as those for cast iron. 

Hodgkinson gives C = 65000 f 
Kondulet " G = 70800 % 
Weisbach « G = 72000 § 
Kankine " G = 30000 to 40000 || 

It is generally assumed that wrought iron will resist about 
two-thirds as much to crushing as to tension, but the experi- 
ments fail to give a very definite ratio. 

59. resistance of steeij to crushing. — Major Wade 
found the following results from experiments upon the several 
samples of cast steel, all of which were cut from the same bar 
and treated as indicated in the table.^f 



Specimen. 


Length. 


Diameter. 


Crushing in 
lbs. per sq. inch. 


Not Hardened 


1.021 
0.995 
1.016 

1.005 


0.400 
0.402 
0.403 

0.405 


198,944 
354,544 
391 985 


Hardened, low temper 

" mean " 


" high " for tools for 
turning hard steel. . 


372,598 



(Old Edition.) 



* Resistance des Materiaux, Morin, p. 95. 

f Vose, Handbook of Railroad Construction, p. 127. 

% Maharts Civil Engineering, p. 97. 

§ Weisbach, Mech. and Eng., vol. i., p. 215. 

jj Rankings Applied Mech , p. 633. 

*[[ Report on Metals for Cannon, p. 258. 



60 



THE RESISTANCE OF MATERIALS. 



chrome steel resists from 160,000 to 195,000 pounds per 
square inch. Indeed Captain Eads says in his report that the 
crushing resistance of chrome steel may be increased to any de- 
sirable amount by the simple addition of chromium. 

60. resistance of wood to crushing. — The resistance 
of wood to crushing depends as much upon its state of dryness, 
and conditions of growth and seasoning, as its tenacity does. 
The following are a few examples : — 



Kiud of Wood. 



Ash 

Oak {Englisli) 
Pine {Pitch).. 




Very Dry. 



9,360 

10,058 

6,790 



These results, compared with the corresponding numbers in 
article 22, show that these kinds of wood will resist from 1J- to 
nearly 2 times as much to tension as to compression. For other 
examples see the Table in the appendix. 

61. RESISTANCE OF GLASS TO CRUSHING. We Owe most 

of our knowledge of the strength of glass to ¥m. Fairbairn and 
T. Tate, Esq. According to their experiments we have the 
following results for the crushing resistance of specimens of 
glass whose heights varied from one to three times their 
diameter. 



MEAN CRUSHING RESISTANCE OF CUT-GLASS CUBES AND ANNEALED 
GLASS CYLINDERS. 



Description of the Glass. 


Weight per Square Inch. 




Cubes. 


Cylinders. 


Flint Glass . 


lbs. 
13,130 

20,206 
21,867 


lbs. 

27,582 




31,876 


Crown Glass 


31,003 




Meat. 


18,401 


30,153 







COMPRESSION. 61 

The ratio of the mean of the resistances is as 1 to 1.6 nearly. 

The cylinders were cnt from round rods of glass, and hence 
retained the outer skin, which is harder than the interior, while 
the cubes were cut from the interior of large specimens. This 
may partially account for the great difference in the two sets of 
experiments. The cubes gave way more gradually than the 
cylinders, but both fractured some time before they entirely 
failed. The cylinders failed very suddenly at last, and were 
divided into very small fragments. The specimens had rubber 
bearings at their ends, so as to produce an uniform pressure 
over the whole section. 

©S. strength of pii^ars. — The strength of pillars for 

incipient flexure has been made the subject of analysis by Euler 
and others, but practical men do not like to rely upon their 
results. Mr. Hodgkinson deduced empirical formulas from ex- 
periments which were made upon pillars of wood, wrought iron, 
and cast iron. The experiments were made at the expense of 
Wm. Fairbairn, and the first report of them was made to the 
Royal Society, by Mr. Hodgkinson, in 1840. The following 
are some of his conclusions : — 

1st. In all long pillars of the same dimensions, when the 
force is applied in the direction of the axis, the strength of one 
which has flat ends is about three times as great as one with 
rounded ends. 

2d. The strength of a pillar with one end rounded and the 
other flat, is an arithmetical mean between the two given in the 
preceding case of the same dimensions. 

3d. The strength of a pillar having both ends firmly fixed, is 
the same as one of half the length with both ends rounded. 

4th. The strength of a pillar is not increased more than ith 
by enlarging it at the middle. 

To determine general formulas, bars of the same length and 
different sections were first used ; then others, having constant 
sections and different lengths ; and formulas were deduced 
from the results. The formulas thus made were compared with 
the results of experiments on bars whose dimensions differed 
from the preceding. The following are the results of some of 
his 



62 



THE KESISTANCE OF MATERIALS. 



EXPERIMENTS ON SQUARE PILLARS. 



Lenerth of 
the bars. 


Side of the square. 


Crushing weight. 


Exponent of the 
side. 


Feet. 

10 
10 

5 


Inches. 

0.766 

1.51 

1.00 

1.50 

1.02 

1.53 

0.50 

1.00 

0.50 

1.00 

0.502 

1.00 

0.502 

0.76 


Lbs. 

1.948 ) 
23,025 ) 

4,225 ) 

23,025 S 

10,236 l 

45,873 ( 

583) 

9,873 j" 

1,411 I 
18,038 J 

4,216 ) 
27,212 \ 

4,216 ) 
15,946 [ 


3.57 
4.17 
3.69 
4.08 
3.67 
2.69 
3.28 






Mean 


3.59 



The fourth column is computed as follows : — 
Suppose that the strengths are as the x power of the diame- 
ters, then for the first bar we have 

/l-Siy _ 23025 

V0.766/ 



or 1.971* = 11.30 



1948 

log. 1.971 
The others are computed in the same way. 
An examination of the table shows that when the square 
section is the same, the strength varies inversely as some func- 
tion of the length. Thus, of two bars whose cross section is one 
square inch, the one five feet long is nearly four times as strong 
as the one ten feet long. 

Let I — length of one, 
V = " of other, 
d = diameter of first one, 
d f = " of the second one, and 
y z= the power of the length. 



Then the strength of the first one is, P = constant x 



d 



8.59 



second is, P' = constant x j t 



P 
d tZM 



COMPRESSION. 



63 



p .^ 3.59 _ V y 
" P'\d) ~\l) 

in which substitute the values from any two experiments. Thus 

if we take from the table 

V = 10 feet, d' = 1 inch, P' = 4225 lbs., and 

I = 5 feet, <# = 1 inch, and P = 18038 lbs., we have 

18038 



4225 



2^ 



y 



_ log. 4.2694 
%. 2 



2.094 



Proceed in a similar way with each of the others and take 
the mean of the results for the power to be used. In this way 
was formed the following 

TABLE 

For the absolute strength of columns. 

in which P = crushing weights in gross tons, 

d = the external diameter, or side of the column in inches, 
c?i = the internal diameter of the hollow in inches, and 
I = the length in feet. 



Kind of Column. 



Both ends rounded, the 
length of the column ex- 
ceeding fifteen times its 
diameter. 



Both ends flat, the length of 

the column exceeding thirty 

times its diameter. 



TONS. 



Solid Cylindrical Columns of 
cast iron. 

Hollow Cylindrical Columns ) 
of cast iron ) 

Solid Cylindrical Columns of [ 
wrought iron ) 

Solid Square Pillar of Dant- 
zic oak 

Solid Square Pillar of red dry 
deal 



P = 14. 



P = 13 



P = 42 



d — "i 



d 3 ' 



TONS. 
,73.66 

P = 44.16 U- 



P = 44.34^— 

6 



P = 133.75 



P = 10.95 ^- 
P = 7.81 % 



The above formulas apply only in cases where the length is 
so great that the column breaks by bending and not by simple 
crushing. If the column be shorter than that given in the 



64 THE RESISTANCE OF MATERIALS. 

table, and more than four or five times its diameter, the strength 
is found by the following formula : 

W=— F - C -K- m) 

in which P — the value given in the preceding table, 

K = the transverse section of the column in square 

inches, 
C = the modulus for crushing in tons (gross) per 

square inch, and 
W = the strength of the column in tons (gross).* 
Experiments have been made upon steel pillars which gave 
good results, f 

63. weight of pillars — From the first formula of the 
preceding table we find 



d 



1491T.W 



The area of the cross-section is J * d 2 , and the volume in 
inches =±£ * d 2 1. 

Cast iron weighs 450 pounds to the cubic foot, hence the 

weight = j^g x 3 x ** x I = m x 3.1416 x .— — 

If P is given in pounds, this coefficient must be divided by 

2240 T -^ 

.-. weight in pounds = 0.0121702 {P ms - Z 3 - b8 )T-H . . .(25) 

If the pillar is hollow the section of the iron is J *r (d? — d 2 ), 
and if n is the ratio of the diameters, so that d x = n d this be- 
comes 

12 
J * d 2 (1 — n 2 ) ; and its volume in inches = — • ?r d 2 (1 — n 2 ) I; 

450 
and its weight in pounds = — — x 3 x *■ d 2 (1 — n 2 ) I. 

* James B. Francis, C. E., has published a set of tables which gives the 
strength of cast-iron columns, of given dimensions, by means of equation (24), 
and also by those in the above table. 

f London Builder, No. 1211. 



COMPRESSION. 



65 



If the value of d from the second equation of the first column 
in the preceding table, be substituted in the preceding equa 
tion, we find the 

weight injpounds = 
25 ?r 1 — n* / \_ i 

32 (2240 x 13 )x.W(l - „..»)t.** l ( P - ? ') TOT W) 

To find the weight for a five-fold security these results must 

be multiplied by Sttss. 

Proceeding in this way with each of the cases given above 
and we form the following : 

TABLE 

Of the weights in pounds of pillars in terms of their lengths (J,) in feet, when 
loaded to one fifth their crushing strength (P) in pounds. 





Weight in pounds. 


Kind of Pillar. 


Both ends rounded. 
1 > 15 a. 


Both ends flat. 
1 > 30 I. 


Solid Cylindrical Col- 
umn of cast iron. 


0.028648953 (P.P- 5 *) r;8¥ 


0.009321706 (P.Z 3 - 475 ) ^^ 


Hollow Cylindrical Col- 


1-n 2 


1-71 3 


umns of cast iron, 
di = nd. 


(1-71 3 - 76 ) l '** 
/p I* .5 8\ iTsa" 


u.uuyjuuiO't ! * 

(1_^3. 55) T-.7T6 
/p^3.4 7 5\7"V7"7"5 


if ?i=0.98 


0.003881655 (P. P' 58 ) r7 * 8 


0.001658133 (P.Z 3 - 475 ) "^ 


if n =0.95 


0.006001775 (P.J 3 - 58 ) *** 


0.002489827 (P.Z 3 - 475 ) T ^ rTS 


if ^=0.925 


0.007265678 (PJ 3 - 68 ) T ^« 


0.002987882 (P. Z 3 " 175 ) 1 - 776 


if 7i=0.90 


0.008396144 (PJ 3 - 58 ) r ~- 88 


0.003406063 (P. 7 /*- 476 ) ^^ 


if ?j=0.875 


0.009373430 (Pi 3 - 58 ) ~ 8l? 


0.003773531 (P.Z 3 - 475 ) J 77 " 7 * 


if 7i=0.85 


0.010261387 (P. Z 3 - 68 ) 1 - 88 


0.004106903 (P.Z 3 - 475 ) T" 77 ^ 


if 71=0.80 


0.011862713 (PJ 3 - 58 ) r ^« 


0.004702651 (P.£ 3 - 47S ) ^^ 


7i=0.75 


0.013297905 (P.Z 3 - 68 ) ^ 8 


0.005233352 (P.Z 3 - 475 ) ^™ 


Solid Cylindrical Col- 
umns of Wrought 
Iron, 


0.014115831 (P.Z 3 - 58 ) *&* 


0.004993604 (P. p^T&rr 


Square Column of Dant- 
zic Oak. 


(Cubic foot weighs 47.24 
pounds.) 


0.001223770 (pfy) 



66 



THE RESISTANCE OF MATERIALS. 



If the thickness of the metal (t) and the external diameter 
are given, n may be found as follows : d — 2t = internal diame- 
ter, hence n = ^-^ = 1 — ^. For instance, if the external 
diameter is 6 inches, and the thickness f of an inch, the inter- 
nal diameter is 54- inches and n = 5£ — 0.875. 

The iron used in the preceding experiments was Low Moor 
No. 2, whose strength in columns is about the mean of a gi eat 
variety of English cast iron, the range being about 15 per cent, 
above and below the values given above. 

64. — condition of the casting. — Slight inequalities in 
the thickness of the castings for pillars does not materially af- 
fect the strength, for, as was found by Mr. Hodgkinson, thin 
castings are much harder than thicker ones, and resist a greater 
crushing force. In one experiment the shell of a hollow column 
resisted about 60 per cent, more per square inch than a solid 
one.* But the excess or deficiencv of thickness should not in 
an}' case exceed 25 per cent, of the average thickness. f Thus, 
if the average thickness is one inch, the thickest part should not 
exceed 1J inch, and the thinnest part should not be less than f 
of an inch. 

It is also found that in large castings the crushing strength 
of the part near the surface does not much exceed that of the 
internal parts. 

©5. COMPRESSION OF TUBES.— BUCK- 
LING. — Wrought iron tubes when sub- 
jected to longitudinal compressive stresses 
may yield by crushing like a block, or by 
bending like a beam, or by buckling. 
The first takes place when the tube is 
very short ; the second, when it is long 
compared with the diameter of the tube ; 
and the last, for some length which it is 
difficult to assign, intermediate between 
the others. 

The appearance of a tube after it has 
yielded to buckling is shown in Figs. 15 
and 16. 

f 8t<mey on Strains, vol. ii. , p. 206. 




Fig. 15. Fig. 16. 



* Phil Trans., 1857, p. 890. 



COMPRESSION. 



67 



The experiments heretofore made do not indicate a specific 
law of resistance to buckling ; but the following general facts 
appear to be established :* — 

1. The resistance to buckling is always less than that to 
crushing ; and is nearly independent of the length. 

2. Cylindrical tubes are strongest ; arid next in order are 
square tubes, and then rectangular ones. 

3. Eectaugular tubes, | ; | , are not as strong as tubes of this 
form Q 3 The tubes in bridges and ships are generally rec- 
tangular or square. 



COLLAPSE OF TUBES. 

60* THE RUPTURE OF TUBES 

which are subjected to great external 
normal pressure is called " a collapse.'' 
The flues of a steam-boiler are sub- 
jected to such an external pressure, 
and in view of the extensive use of 
steam power, the subject is very im- 
portant. The true laws of resistance 
to collapsing were unknown until the 
subject was investigated by Win. Fair- 
bairn. Experiments were carefully 
made, and the results discussed by him 
with that scientific ability for which 
lie is so noted. They were published 
in the Transactions of the Royal 
Society, 1858, and republished in his 
" Useful Information for Engineers," 
second series, page 1. 

The tubes were closed at each end 
and placed in a strong cylindrical ves- 
sel made for the purpose, into which 
water was forced by a hydraulic press, 
thus enabling him to cause any desir- 
able pressure upon the outside of the 
tube. In order to place the tube as 
nearly as possible in the condition of 




Fig. 18. 



Fig. 17. 



Civ. Eng. and Arch. Jour., vol. xxviii., p. 28. 



68 THE BESISTANCE OF MATERIALS. 

a flue in a steam-boiler, a pipe which communicated with the 
external air was inserted into one end of the tube. This pipe 
permitted the air to escape from the tube during collapse. 

The vessel, pipe, tube, and their connections were made 
practically water-tight, and the pressure indicated by gauges. 

Fig. 17 shows the appearance and cross-section at the middle 
of the short tubes after the collapse ; and Fig. 18 of a long 
one. Although no two tubes appeared exactly alike after the 
collapse, yet the examples which I have selected are good types 
of the appearances of thirty tubes used in the experiments. 

The tubes in all cases collapsed suddenly, causing a loud 
report. In the first and second tubes the ends were supported 
by a rigid rod, so as to prevent their approaching each other 
when the sides were compressed. 

The following tables give the results of the experiments : — 



COMPRESSION. 



TABLE I. 







Thickness 






Pressure of 




Product of 






of Plate, 


Diameter 


Length in 


Collapse, lbs. 


Pressure 


the Pressure, 


Mark. 


No. 


inches. 
t. 


in inches. 
d. 


inches. 
L. 


pr. sq. in. of 

Surface. 

P. 


and Length. 
P. L. 


Length, and 
Diameter. 

p. l. a. = p. 


A 


1 


0.043 


4 


19 


170 


3230 




B 


2 


a 




19 


137 


2603 


10412 


C 


3 


u 




40 


65 


2600 


10400 


D 


4 


u 




38 


65 


2470 


9880 


E 


5 


t< 




60 


43 


2580 


10320 


F 


6 


It 




60 


140* 


2800 






Mean 


2714 


10253 


G 


7 


a 


6 


30 


48f 


1440 




H 


8 


u 




29 


47+ 


1263 




J 


9 


u 




59 


32 


1888 


11328 


K 


10 


u 




30 


52 


1560 


9360 


L 


11 


it 




30 


65 


1950 


11700 


M 


12 


u 




30 


85t 


? 






Mean 


1620 


10796 


N 


13 


a 


8 


30 


39 


1170 


9360 





14 


a 


u 


39 


32 


1248 


9984 


P 


15 


a 


(< 


40 


31 


1240 


9920 




Mean 


1219 


9754 


Q 


16 


u 


10 


50 


19 


950 


9500 


R 


17 


u 


c« 


30 


33 


990 


9900 




Mean 


970 


9700 


S 


18 


a 


12.2 


58* 


11.0 


643.7 


7850 


T 


19 


u 


12 


60 


12.5 


750 


9000 


V 


20 


u 


t< 


30 


22 


662 


7920 




Mean 


685.2 


8256 



* This tube had two solid rings soldered to it, 
three tubes, as shown in Fig. 19. 



inches apart, thus practically reducing it to 




Fig. 19. 



t The ends of both were fractured, causing collapse, perhaps before the outer shell had 
tained its maximum. 
X A tin ring had been left In by mistake, thus causing increased resistance to collapsing. 



70 THE KESISTANCE OF MATEEIALS. 

67. discussion of results. — By comparing the tubes of 
the same diameter and thickness, but of different lengths, we 
see that the long tubes resist less than the short ones ; hence, 
the strength is an inverse function of the length, and an ex- 
amination of the seventh column shows that it is nearly a sim- 
ple inverse function of the length. The first of the 4 inch 
tubes is so much, stronger than the others, it may be neglected 
in determining the law of resistance, although it differs from a 
mean of all the others by less than \ of the mean. An exami- 
nation of the several cases indicates that we may safely assume 
that the resistance to collapsing varies inversely as the lengths 
of the lubes* 

The mean of the results for the several diameters in the 
last column shows that the resistance diminishes somewhat 
more rapidly than the diameter increases ; but this includes the 
error, if any, of the preceding hypothesis. As the power of 
the diameter is but little more than unity, it seems safer to con- 
clude, for all tubes less than 12 inches in diameter, as Fair- 
bairn does, that the resistance of tithes to collapsing varies in- 
versely as their diameters, 

68. law of thickness. — Experiments were also made 
to determine the law of resistance in respect to the thickness. 
Comparatively few experiments were made of this character, 
but these few gave remarkably uniform results. One of the 

* A more exact law may be found as follows : — Let P = the compressing 
force per square inch ; G = a constant for any particular diameter and 
thickness, I = the length, and n the unknown power. Then 

P = -^r— for one case. 

C 
Pi — yv ^ or another. 

6] 

log. -p- 

h 

log. i 

By means of this equation, and any two experiments in which the thickness 
and diameter are the same, n may be found, and by using several experiments 
a series of values may be found from which the most probable result can be 
obtained. But in this case the mean result is so near unity, there is no prac- 
tical advantage secured by finding it. 



COMPKESSION. 



71 



tubes (No. 24), was made with a butt joint, as shown in Fig 
20, and the others with lap joints, as in Fig. 21. 





Fig. 20. Fig. 21. 

The following are the results of the experiments: 

TABLE II. 







Thickness. 


Diameter. 


Length in 


Pressure per 


1 Product. 


Mark. 


No. 


t. 


d. 


inches. 


square inch. 


PL. \ L. P. d. 










L. 


P. 


1 =P. 


w 


21 


0.25 


9 


37 


(450) 


TJncollapsed. 


X 


22 


0.25 


18£ 


61 


420 


25620 


480375 


Y 


23 


0.14 


9 


37 


262 


9694 


89046 


Z 


24 


0.14 


9 


37 


378 


13986 


125874 


JJ 


33 


0.125 


14^ 


60 


125 


7500 


108750 



Tubes Nos. 23 and 24 were exactly alike in every respect 
except their joints ; and it appears that the butt joint, No. 24, 
is 1.41 times as strong as the lap joint, a gain of 41 per cent. 
But this is a larger gain than is indicated in other cases ; for 
instance, No. 33, which is also a lap joint, offers a greater re- 
sistance as indicated in the last column, than No. 23, although 
the former is not as thick as the latter. Still it seems evident 
that butt joints are stronger than lap joints, for with the 
former the tubes can be made circular, and there is no cross 
strain on the rivets, conditions which are not realized in the 
latter. 

The resistance of the 23d is so small compared with others, 
it is rejected in the analysis. 

We observe that the resistance varies as some power of the 
thickness ; if then C and n be two constants to be determined 
by experiment, and we use the notation given above, we shall 
have for the pressure of collapse of one tube, 



72 THE RESISTANCE OF MATERIALS. 

rtfn 

P = ^x r - dPL = p= Cfl ( 27 > 

and for another tube 

pfn 

P '^Tl '"■ * p * z ^=^= a " ( 28 ) 

Hence we have 

P = /* \" 

"—^kt » 

mio =f=i- < s °) 

TO FIND THE CONSTANTS 71 AND C. 

The mean of the mean of the valnes of p from Table I. is 
p = i [10253 + 10796 + 9754 + 9700 + 8256] = 9752 and 
t = 0.043. 

Using these values and others taken from the preceding 
tables, and the following values may be found for n : — 

In equation (29) makej? = 4S0375, t = 0.25, p x = 9752, t, 
= 0.043 ; and we get 

log. 480375 - % 9752 
%. 0.25 - %. 0.043 ' ' 

Similarly, taking p = 480375, t == 0.25, ^ = 10253, f, = 
0.043 ; and we get 

%. 480375 - %. 10253 _ 
W " %. 0.25 - log. 0.043 * * 

The mean value of p for all but the 12-inch tubes in Table 
Lis 

p = i (10253 + 10796 + 9754 + 9700) = 10125 ; 
hence, using p = 125874, t = 0.14,^ = 10125, t x = 0.043; 
and we set 

_ fog. 125874 - log. 10125 _ . 

%. 0.14 - %. 0.043 
and taking jp = 108750, t = 0.125,^ = 10125 and t x = 0.043; 
we get 

_ log. 108750 - log. 10125 _ 2Q3 
71 ~~ %. 0."125 - %. 0.043 "" ' ' 
an 1 the mean of these results is, n — 2.18. 



to* 

n 



COMPRESSION. 73 

Fairbaim made it 2.19 by including some data which I have 
rejected as paradoxical ; I have also given more weight to those 
cases which gave nearly uniform results. The difference, how- 
ever, of 0.01 is too small to seriously affect practical results. 

To determine the constant, O, substitute the proper values 
taken from the preceding tables in equation (30), and we have 
for four cases the following : — 

G =S* =9,298,900. 

• C= ggg = 9,864,300. 

tf=gj£ = 9,144,000. 

C>= ^ = 10,109,400. 

The mean of which is C - 9,604,150. Calling G = 
9,600,000 and equation (27) becomes : 

P = 9,600,000 ~ (31) 

Cu.Ju 

If L be given in feet, so that L — 12 Lj, we have 
P = 800,000 |~ (32) 

The coefficient, 9,600,000, applies only to the kind of iron 
used; but the exponent, 2.18, is supposed to be constant for all 
kinds of iron. 

69. FORMULA FOR THICKNESS TO RESIST COLLAPSING. 

— Equation (31) readily gives the following expression for find- 
ing the thickness in inches of a tube to resist collapsing : — 

log. 100 t = l0g " P t jf {d - L) - 1.203 (33) 

.2.18 

70. elliptical tubes. — Experiments made upon ellipti- 
cal tubes showed that the preceding formula would give the 
strength, if the diameter of the circle of curvature at the ex- 
tremity of the minor axis is substituted for d. The diametei 



74 THE RESISTANCE OF MATERIALS. 

2$ 5 
of curvature is —j-, in which a is the major and h the minor 

axis. 

Experiments made upon tubes in which the ends were not 
connected by internal rods, showed that the resistance was in- 
versely as their length. 

71. very long tubes. — Some experiments were made 
upon a tube 35 feet long and one 25 feet long. Sufficient pres- 
sure was applied to distort them, but not to collapse them, and 
it was found that Equation (31) erred by at least 20 per cent., 
giving too small an amount. It was, however, very evident 
that the length was still a very important element in the 
strength. 

72. COMPARISON OF STRENGTH FROM EXTERNAL. AND 

internal presslre. — Let p be the internal pressure per 
square inch at which the tube is ruptured, then for tubes of the 
same thickness and diameter we have from Equations (18) and 
(32), by calling T = 30,000 lbs., 

P 13.33 t 1 - 18 

Ifp= P,thenZ = 13.33 t 1 - 18 . 

If t = 0.25, then we find L = 2.59 feet, that is, a tube whose 
thickness is J of an inch, and whose length is 2.59 feet, is 
equally strong whether subjected to internal or external pres- 
sure. 

If the tube is so thick that the unequal stretching of the 
fibres must be considered, then Equation (20) must be compared 
with Equation (32), in which case we have : — 

p = T d.L f 

T 800,000 X (r + t) t 1 ' 18 

Up = P, T= 40,000 lbs., and 2r = d = 4 inches ; 
then 2* 1 - 18 + f- 18 = | L f . 

If t = i inch, L — 5.504 feet. 
If t = 1 " L = 15.000 feet. 

73. RESISTANCE OF GLASS GLOBES TO COLLAPSING. — 

Fairbairn also determined that glass globes and cylinders fol 



C0MPBESSI01T. 75 

l>wed the same general law of resistance. For globes of flint 
glass he found : 

i>, =28,300,000 Jj( (34) 

and for cylinders of flint glass : 

P x = 740,000-^ (35) 

providing that their length is not less than twice, nor more than 
six times their diameter. Dividing Equation (35) by (31) gives 
P, _ 0.0770 
P ~ f-™ ' 

P 

If £=0.04:3 in., -=■* = 0.896 ; or the glass cylinder is nearly ^ 

as strong as the iron one. If they are equally strong, P = P x 
.-. t - 0.0373 of an inch. 



76 



THE RESISTANCE OF MATERIALS. 



CHAPTEK III. 



THEOEIES OF FLEXURE AND RUPTURE FROM TRANSVERSE 

STRESS. 

74. — remark. — The ancients seem to have been entirely 
ignorant of the laws which pertain to the resistance of solid 
bodies. They made some rude experiments to determine the 
absolute strength of some solids, especially of stone. They may 
have recognized some general facts in regard to the strength of 
beams, such as that a beam is stronger with its broad side ver- 
tical than with its narrow side vertical, but we find no trace of 
any law which was recognized by them. This department of 
science belongs wholly to modern times. A very brief sketch 
of the history of its development is given below.* 

The greater part of this chapter will be much better under- 
stood after reading Chapters. IY, Y, and YI. 

75. — GAiiiiiEO's theory. — Galileo was the first writer, of 
whom we have any knowledge, who endeavored to establish the 
mathematical laws which govern the strength of beams, f He- 
assumed — 

1st. That none of the fibres were elongated or compressed. 

2d. When a beam is fixed at 
one end, and loaded at the other, 
it breaks by turning about its 
lower edge, Fig. 22, or if it be 
supported at its ends and loaded 
at the middle of the length, it 
would turn about the upper edge ; 
hence every fibre would resist ten- 
sion. 

3. Every fibre acts with equal 
energy. From these he readily 

Resistant des Corps 




Fig. 22. 



* For a more complete history, see introduction to 
Solides," par Navier. 3d edition. Paris, 1864. 
+ Opere di Galileo. Bologne, 1656. 



FLEXURE AND EUPTTJEE FE0M TEANSVEESE STEESS. 77 

deduced, — that, when one end is firmly fixed in a wall or othei 
immovable mass, the moment of resistance of the section^ equal 
to the sum of all the fibres, or the transverse section, multiplied 
by the resistance of a unit of section, multiplied by the distance 
of the centre of gravity from the lower edge. Hence, in a rec- 
tangular beam, if 

T = the tenacity of the material, 

h = the breadth, and 

d = the depth of the beam ; 

the moment of resistance is 

Tbd x id = i Tbd 2 (34) 

76. — Robert hooke's theory. — Kobert Hooke was one 
of the first, and probably the first, to recognize the compressi- 
bility of solids when under pressure. In 1678 he announced 
his famous principle, Ut tensio sic vis / which he gave in an 
anagram in 1676, and stated as the basis of the theory of elasti- 
city that the extensions or contractions were proportional to the 
forces which produce them, and also that when a bar was bent 
Jie material was compressed on the concave side and extended 
on the convex side. 

77. — MARIOTTE'S AND LEIBNITZ'S THEORY. — Mariotte, 

in 1680, investigated the subject, and finally stated the follow- 
ing principles : — 

1st. The material is extended on the convex side and com- 
pressed on the concave side. 

2d. In solid rectangular sections the line of invariable fibres 
(or neutral axis) is at half the depth of the section. 

3d. The elongations or compressions increase as their distance 
from the neutral axis. 

4th. The resistance is the same whether the neutral axis is at 
the middle of the depth or at any other point. 

5th. The lever arm of the resistance is f of the depth. 

We here find some of the essential principles of the resist- 
ance to flexure, as recognized at the present day ; but the two 
last are erroneous. As hereafter shown, the neutral axis is at 
half the depth, and the lever arm is f of \ the depth. 

Leibnitz's theory, given in 1684, was the same as Mariotte's. 



78 THE RESISTANCE OF MATERIALS. 

78. — james Bernoulli's THEORY was essentially the 
same as Mariotte's, except that he stated that extensions and com- 
pressions were not proportional to the stresses. " For," said he, 
" if it is true, a bar might be compressed to nothing with a 
finite force." On this point see Article 16. He was the first to 
give a correct expression for the equation of the elastic curve. 

79. parent's theory. — Parent, a French academician 
of great merit, but of comparatively little renown, published, in 
1713, as the result of his labors, the following principles, in 
addition to those of his predecessors : — 

1st. The total resistance of the compressed fibres equals the 
total resistance of the extended fibres. 

2d. The origin of the moments of resistance should be on 
the neutral axis. 

By the former of these principles the position of the neutral 
axis may be found, when the straining force is normal to the 
axis of the beam ; and by the latter he corrected the error of 
Mariotte and Leibnitz ; showing that the ratio of the absolute to 
the relative strength is as six times the length to the depth 
instead of three, as will be shown hereafter. 

8®. coulomb, in 1773, publisheb the most scientific 
work on the subject of the stability of structures which had 
appeared up to his time. He deduced his principles from the 
fundamental equations of statics, and generalized the first of 
the principles of Parent, which is given above, by saying that 
the algebraic sum of all the forces must be zero on the three 
rectangular axes. This establishes the position of the neutral 
axis when the applied forces are oblique to it, as well as when 
they are normal. He also remarked, that if the proportionality 
of the compressions and extensions do not remain to the last, or 
to the point of rupture, the final neutral axis will not be at the 
centre of the section. 

8 1 . modulus of elasticity — In 1807 Thomas Youngs 
introduced the term modulus of elasticity, which we have de 
fined as the coefficient of elasticity in Article 5. After this 
several writers, among them Duhamel, Javier in his early 
writings, and Barlow in his first work, stated the erroneous prin- 



FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 79 

ciple, that the sum of the moments of the resistances to compres- 
sion equalled those for tension. 

83. in 1824 navier PUBLISHED the lectures which he 
had given to V&ole des Ponts et Chaussees, in which he estab- 
lished more clearly those principles of elastic resistance, and 
resistance to rupture, which have since his day been accepted 
by nearly all writers. He was the first to show that when the 
stress is perpendicular to the axis of the beam, the neutral axis 
passes through the centre of gravity of the transverse sections. 
His most important modifications in the analysis was in making 
ds=dx, or otherwise, considering that for small deflections the 
tangent of the angle which the neutral axis makes with the 
original axis of the beam is so small compared with unity that 
it may be neglected / and also, that the lever arm of the force 
remains constant during flexure. These principles we have 
used in Chapter Y. He resolved many problems not before 
attempted, and became an eminent author in this department 
of science. 

83. the common theok¥. — The theories of flexure and 
of rupture which result from these numerous investigations, I 
will call, for convenience, the common theory. It consists of 
the following hypotheses : — 

1st. The fibres on the convex side are extended, and on the 
concave side are compressed, and there are no strains but com- 
pression and extension. 

2d. Between the extended and compressed fibres (or elements) 
there is a surface which is neither extended nor compressed, but 
retains its original length, and which is called the neutral sur- 
face, or in reference to a plane of fibres it is called the neutral 
axis. 

3d. The strains are proportional to their distance from the 
neutral axis. 

4th. The transverse sections which were normal to the neutral 
axis of the beam before flexure, remain normal to the neutral 
axis during flexure. 

5th. A beam will rupture either by compression or extension 
when the modulus of rupture is reached. 

6th. The modidus of rupture is the strain at the instant of 



80 THE RESISTANCE OF MATERIALS. 

rupture upon a unit of the section which is most remote from 
the neutral axis on the side which first ruptures. This is 
called R. 

It is found that this theory does not conform well with the 
results of experiment.* For instance, if a cast-iron beam be 
supported at its ends, and broken by a weight placed at the 
middle, it appears from the theory above given that the beam 
would break when the strain (R) on the extreme fibres equals 
the value of the tenacity (T) of the metal — or 16,000 lbs. 
(See page 58.) But the value of R as found from the formula, R 

PI 

= § j~ which is deduced in accordance with the above theory, 

and is given in Chapter VI., is about 35,000 lbs. (See the table 
in the Appendix.) This value is less than the crushing strength, 
6 Y , of the metal — or 96,000 lbs. (See page 58). Hence the 
value of R is nearly 2J times that of T, and more than -J- that 
of G. 



Again, we have for Ash 



T = 17,200 pounds 

0= 9,000 " 
R = 12,000 " 
R = H <7andf T nearly. 



* Moslems Mech. and Arch., p. 557. " The elasticity of the material has 
been supposed to be perfect up to the instant of rupture, but the extreme fibres 
are strained much beyond their elastic limits before rupture takes place, while 
the fibres near the neutral axis are but slightly strained, and hence the law of 
proportionality is not maintained, and the position of the neutral axis is 

changed, and the sum of the moments is not accurately -=- (see equation 170). 

To determine the influence of these modifications we must fall back upon ex- 
periment, and it has been found in the case of rectangular beams that the error 

will be corrected if we take — T (= M) instead of T. where m is a constant 

depending upon the material. " 

Weisbach, vol. ii., 4th ed., p. 68, foot-note, says, "Excepting as exhibiting 
approximately the laws of the phenomena, the theory of the strength of mate- 
rials has many practical defects. " 

Maj. Wade, in his Report to the Ordnance Department, p. 1, says: — "A trial 
was made with cylindrical bars in place of square ones. These generally 
broke at a point distant from that pressed, and the results were so anomalous 
that the use of them was soon abandoned. The formula by which the strength 
of round bars is computed appears to be not quite correct, for the unit of 
strength in the round bars is uniformly much higher than in the square bars 
cast from the same iron." 



FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 81 

A similar result is found for other materials. Hence gener- 
ally, the value of R for any given material is between those of 
T and O, but there is no known relation between them which 
would enable us to determine the value of one from the other 
two. The values of R in the tables were deduced from experi- 
ments upon rectangular beams, as will hereafter be shown; 
and hence, if the common theory is correct, R should equal the 
value of the lesser resistance, whether it be for compression or 
extension ; but it does not. This discrepancy between theory 
and the results of experiment, led Mr. Barlow to investigate the 
subject farther, and it resulted in a new theory which he calls 
"Resistance to Flexure" — an expression which I consider un- 
fortunate, as it does not express his idea. "Longitudinal 
Shearing" would express his idea better, as will appear from 
the following article : — 

84. barlows theory. — According to the common theory 
the resistance at a section is the same as if the fibres acted in- 
dependently of each other, and the transverse section remained 
normal to the neutral axis. But Barlow supposed that in order 
to keep the transverse sections normal to the neutral axis, 
the consecutive longitudinal planes of fibres must slide over 
each other, and to this movement they offer a resistance. (This 
point is discussed in Chapter IX.) 

He presented his view to the Royal Society (Eng.), in 1855, 
and it has since been published in the Civil Engineer and 
Architects Journal, vol. xix., p. 9, and vol. xxi., p. 111.* The 
subject is there discussed in a very able and thorough manner, 
and although he may have failed to establish his theory, yet the 
results of his, analysis seem to agree more nearly with the re- 
sults of experiment than those obtained by any other theory 
heretofore proposed. 

It is admitted in this theory that a beam will rupture when 
the stress upon any fibre equals its tenacity, or its resistance to 
compression, as the case may be. But, on the other hand, when 
the adjacent fibres are unequally strained, as they are in the 
case of flexure, it requires a greater stress to produce this 

* Civ. Eng. and Arch. Jour., vol. xix., p. 9, Barlow says that the strength 
of a cast-iron rectangular bar, as found from existing theory, cannot be recon- 



82 



THE KESISTANCE OF MATERIALS. 




strain than it would if the fibres acted independently, according 
to the previously assumed law. This, Barlow 
makes evident from the following example : — 
If a weight P, Fig. 23, is suspended on 
a prismatic bar, BCEF, all the fibres will 
be equally strained, and hence equally elon- 
gated. 

But if the bar ABOD be substituted for 
the former, and the weight P acts upon a part 
of the section, as shown in the figure, it is evi- 
dent that all the fibres will not be equally 
strained, and hence will not be equally elon- 
gated ; and if the force P was just sufficient to 
rupture the bar FB OF, it will not be sufficient 
to rupture the bar ABOD, although P acts 
directly upon the same section, for the cohe- 
Fig. 23. sion of the particles along FF will not permit 

the fibres next to that line to be elongated as much as if the 
part AFED were removed; and these fibres will act upon 
those adjacent, and so on, until they produce an effect upon B O. 
From this we see that it takes a greater weight than P acting 
upon the section FO 'to produce a strain T~per unit of section, 
when the part ADFFis added. It is also evident that if the 
section of ABOD is twice as great as FBOF, it will not take 
twice P to rupture the fibres on the side BO. 

A phenomenon similar to this takes place in transverse 
strain. One side is compressed and the other elongated ; and 
the fibres less strained aid those which are more strained by 
virtue of the cohesion which exists between them, and it takes 
a greater load to cause a strain, T, longitudinally upon the 
fibres on the convex side, or of O upon those on the concave 
side, than it would if there were no cohesion between the hori 
zontal laminpe. 

ciled with the results of experiment if the neutral axis be at the centre of the 

sections. He then proceeded to show by experiment that the neutral axis is 

at the centre, and then remarked that the formula commonly used for a beam 

2 T bd- 
supported at the ends and loaded in the middle, or P = d - — ~ did not give 

half the actual strength if T is the tenacity of the iron. He then proceeds 
to point out a new element of strength, which he calls "Resistance to 
Flexure." 



FXEXURE and rupture from transverse stress 83 

his may be illustrated by a pile of boards, Fig. 24. Sup- 




Fig. 24. 

j /se iaat the boards are very thin and perfectly smooth. When 
they ucr> bent they will retain their original length, and mil 
project past each other as shown in the figure. Also if before 
they w-jre deflected straight lines were drawn with a pencil or 
otherwise perpendicularly across the pile, and then the whole 
deflected, it will be found that the lines will not remain contin- 
uous but will be broken. If now there be considerable friction 
between the boards, those on the concave side will be com- 
pressed, and those on the convex side will be elongated ; and 
the cross lines will be more nearly continuous than before. 
Still more, if the successive layers be infinitely thin and held 
together by cohesion, the elements on the concave side will be 
still more compressed and those on the convex more extended 
than they were in the former case, and the cross lines will 
remain straight and normal to the neutral axis, as shown 
in Fig. 25. 




Fig. 25. 

There is, then, at the time of the rupture of a beam, accord- 
ing to this theory, a tensile strain on the extended fibres, and 
a compressive strain on the other fibres, and a longitudinal 
shearing strain between the fibres, due to cohesion. "We think, 
however, that too much importance is given to the longitudinal 
shearing as an element of strength. 

Barlow's Theory consists of the following hypotheses : — 



84 THE RESISTANCE OF MATERIALS. 

1st. The fibres or elements on the convex side are extended, 
and on the concave side compressed. 

2d. There is a neutral surface, as in the common theory. 

3d. The tensile and compressive strains on the fibres are pro- 
portional to their distances from the neutral axis. 

4th. That in addition to these there is a " Resistance to 
flexure " or longitudinal shearing strain, which consists of the 
following principles : — 

a. It is a strain in addition to the direct extensive and com- 
pressive forces, and is due to the lateral cohesion of the adjacent 
surfaces of fibres or particles, and to the elastic reaction which 
ensues when they are unequally strained. 

b. It is evenly distributed over the surface, and consequently 
within the limits of its operation its centre of action will be at 
the centre of gravity of the compressed or of the extended 
section. This force for solid beams Barlow calls <f>, and for f 
or X sections, or open-built beams, it is easily deduced from the 
following principle : — 

c. It is proportional to and varies with the inequality of 
strain between the fibres nearest the neutral axis and those 
most remote. 

From this it appears that if d' is the depth of the horizontal 

flanges of the X section, and d x the distance of the most remote 

fibre from the neutral axis, then the resistance to flexure of the 

d' 
flanges will be <$> — and similarly for other forms. 

5. Sections remain normal to the neutral axis during flexure. 

6. Rupture of solid beams takes place when the strain on a 
unit of section is T + <j>, or C + <j>, whichever is smaller, or 
rather, whichever value is first reached. 

Prof. Barlow made no effort to show the value of the elastic 
resistance of longitudinal shearing in a beam under flexure. 
The effect of this resistance in the flexure of beams will be 
noticed hereafter. 

85. transverse elasticity. — If a beam were destitute 
of elasticity it could not be bent. If it had longitudinal elas- 
ticity only, it could be bent by causing the fibres on the convex 
side to be elongated and those on the concave side to be shortened, 
as explained in the previous articles. If it had no longitudinal 



FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 



85 



elasticity, but a transverse elasticity, it could be bent by forcing 
the successive material sections past each other. Let AB, Fig. 
26,frepresent a beam which is supported at its ends, and which 



m 



Fig. 26. 

is supposed to consist of a succession of perfectly nonelastic 
parts, as a, b, c, etc., and that these parts are joined by infinitely 
thin elastic pieces. If a weight P be placed upon the beam, it 
will cause a deflection similar to that shown in the figure, 
excepting that the visible effect is greatly exaggerated ; but the 
successive sections will set past each other a small amount. If 
now we suppose that the transverse elasticity is uniform and 
continuous from end to end, it seems evident that the deflection 
will take the form of that shown in Fig. 27, in which the effect 




Fig. 27. 

due to the elongation and compression of the fibres is supposed 
to be entirely eliminated. In this case the upper and lower 
sides are straight from the centre to the ends, but they form an 
angle with each other at the centre. 

It will be shown in Chapter IX, that for amorphous bodies 
(called isotropes), in which the elasticity is the same in all di- 
rections, that the coefficient of the transverse elasticity is f- of 
the coefficient of longitudinal elasticity, or G = % E. Such 
bodies, however, are more ideal than real. The elasticity is 
generally different in different directions.* 



See pp. 16 and 17. 



86 THE RESISTANCE OF MATERIALS. 

86, remarks upon the theories. — For scientific pur 
poses it is desirable to determine the correct theory of the 
strength of beams, but the phenomena are so complex that it is 
not probable that a single general theory can be found which 
will be applicable to all the irregular forms of beams used in 
practice. Although Barlow's theory appears plausible, yet ac- 
cording to principle c the resistance to flexure, </>, cannot be 
uniform over the surface, as stated in principle b, because the 
proportionality of the elongations and compressions do not con- 
tinue up to the point of rupture. The common theory is faulty 
beyond what has already been said, in the I section ; for in the 
upper and lower portions the strains on all the fibres are 
not proportional to their distances from the neutral axis, to 
realize which the material should be continuous from the neu- 
tral axis to the remotest fibres. And Barlow's theory is defec- 
tive in the same case, on account of the peculiar strains upon 
the fibres at the angles where the parts join. For rupture, 
then, we can use these theories to ascertain general facts, and 
make the results safe in practice by using a proper coefficient 
of safety ; but for flexure the common theory is approximately 
exact if the elastic limit is not passed, and this is fortunate, for 
the conditions of stability should be founded upon the elastic 
properties rather than on the ultimate strength of the material. 
For the rupture of rectangular beams the common theory will 
be sufficiently exact if the value of R is used instead of T or C 
in the formulas. 



POSITION OF THE NEUTRAL AXIS. 

87. position found experimentally. — According to 
Galileo's, Mariotte's, and Leibnitz's theories, the neutral axis is 
on the surface opposite the side of rupture. 

Professor Barlow made the following experiments : — He took 
a cast-iron beam and drilled holes in its sides, into which were 
fitted iron pins. He carefully measured the distance between 
the pins, before and after flexure, by means of a micrometer, 
and thus found that in solid cast-iron beams bent by a normal 
pressure the neutral axis passes through the centre of the sec- 
tions {Civ. Eng. and Arch. Jour., vol. xix., p. 10). He alsr 



FLEXURE AND RUPTURE FROM TRANSVERSE STR1SS. 8 7 

made the same kind of an experiment on a solid rectangulai 
wrought-iron beam, and with the same result {Civ. Eng. and 
Arch. Jour., vol. xxi., p. 115). 

Some years previous to the preceding experiments, he took a 
bar of malleable iron and cut a transverse groove in one side, 
into which he nicely fitted a rectangular key. When it was 
bent, the fibres on the concave side were compressed, and the 
groove made narrower, so that the key would no longer pass 
through, and thus he showed that the neutral axis was between 
i and -§■ the depth of the beam from the compressed side (Bar- 
low's Strength of Materials, p. 330 ; Jour. Frank. Inst., voL 
xvi., 2d series, p. 194). 

Experiments made at the Conservatoire des Arts et Metiers, 
in 1856, on double T sections, show that it passes through the 
centre of the sections (Morin, Resistance des Materiaux, p. 
137). And experiments made at the same time on rectangulai 
wooden beams showed that it passed at or very near the centre 
of gravity of the sections. 

In these experiments the elasticity of the material was not 
seriously damaged by the strains. To render them complete, 
the strains should have been carried as near to the point of rup- 
ture as possible. 

Louis Nickerson, C. E., of St. Louis, made some experiments 
upon glass by means of polarized light, from which he deduced the 
following as applicable to that and similar amorphous bodies : — 

The neutral axis — as exhibited by polarized light, — from the 
cohesion of material or other cause is extended to a breadth, 
and cannot become a true line until, in reference to the cohe- 
sion, the tensile and compressive forces are infinite. Also that 
its longitudinal direction, like the direction of lines of strain, is 
not an arbitrary one, but resultant from the relative qualities 
and quantities of all the forces in the beam — its evident place 
in physics being that of still water between opposing eddies or 
vortices. 

Results obtained showed that the neutral axis is a flexible 
line, or plane, truly parallel to the top and bottom sides of the 
rectangular beam and passing through the centres of gravity of 
its sections only when the load is evenly distributed from end 
to end, or when the beam is infinitely long, and that, when 



88 THE RESISTANCE OF MATERIALS. 

there is a local pressure, the neutral axis is more or less gov- 
erned in its direction and form by the strain passing from the 
point of local pressure towards the points of support. 

88. POSITION DETERMINED ANALYTICALLY. We knOW 

from statics that the algebraic sum of all the forces on each of 
the rectangular axes must be zero for equilibrium ; hence, if the 
deflecting forces are normal to the axis of the beam, the sum of 
the resistances to compression must equal those for tension. 

1st. Suppose that the coefficient of elasticity for compression 
equals that for tension. Then will the compressions and exten- 
sions be equal at equal distances from the neutral axis. In 
Fig. 28, let R c be the strain on a unit of fibres most remote 
from the neutral axis on the compressed side, and d c = the dis- 
tance of the most remote fibre on the same side ; then, 

-4 = 8 = strain at a unit's distance from the neutral axis. 
d c 

Let k l9 \, k s , &c, be the sections of fibres on one side of the 
neutral axis, at distances of 
2/u Vv V%i & c -? from the axis, and 

h\ li\ h"\ &c, and y\ y", y"\ &c, corresponding quan- 
tities on the other side. 

Then s (k,y x -f-% 2 +%,+&c.) = s (k'y' +k"y" ' +h"'y m + &c), 
or, % 1 +^ 2 +%s + &c.-(7^y4-^y / 4-^ / y // + &c.) - 0, 

or, 2% = (35) 

or the neutral axis passes through the centre of gravity of the 
sections.* 

If the resistance to compression is greater than for tension, 
the neutral axis will be nearer the compressed side than when 
they are equal. 

2. Suppose that the coefficient of elasticity is not the same 
for tension as for compression. 

* The analytical expression for the ordinate to the centre of gravity is 

_ _ k 1 y 1 + *#, + &c k'y' + k"y" + &c, Jjy* V**' 

* - kl + Jc % + &o. + h' + k"+ &c. or * ~ j ydx 

liffydydx = 0, F =0. 



FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 



89 



Suppose that the sections 



Let Fig. 28 represent the beam. 
CM and EF were parallel 
before deflection. If through M.M 




Eig. 28. 



iT, the point where EF in- 
tersects the neutral axis, KH 
is drawn parallel to CM, the 
ordinates between EF and 
FIT will represent the elon- 
gations on one side, and the 
compressions on the other, 
for those fibres whose origi- 
nal length was ZJV. 
Let I = LN, 

x = Ice = the elongation of a fibre at k • 
j? = a pulling or pushing force which would produce A ; 
y = JS"k = distance of any fibre from the neutral axis ; 
h = section of any fibre ; 
E t = coefficient of elasticity for tension ; and 
E c = " " " compression. 

From equation (3) we have, 

i> = f± (36) 

But a is directly proportional to its distance from the neutral 
hence, if c be a constant quantity, whose value is or is not 



axis 



known, we shall have a = cy* 



P = 



__ cEJcy 



I 



Or, if we adopt the same notation as in the preceding case, we 
shall have for the total force tending to produce extension, 



eE 



(37) 



Similarly for compression 
&E 







+ &c.) (38) 

Placing these equal to each other and we have, 

A (%,+%* +%3 + &c) = E c (k>y'+k"y"+k'"y'" +&<>■) 

or, in the language of the integral calculus, 

y o 

E t ?> ydydx = E c s_ y ydydx, (39) 



rite 

* Comparing this equation with equation (45) gives c — — 

P 



90 



THE EESISTANCE OF MATERIALS. 



in which y is an ordinate and x an abscissa. Equation (39) 
enables us to find the position when the form of section is 
known. In most cases, however, the reduction is not easily 
made. 

Example. — Suppose the sections axe rectangular. 

4 h c 

d 



:e 



Let b = AC, 
d = AB, 

?L = a, and 

E e 

y — AE for the superior limit. 
Then equation (39) becomes 

1) 

ydydx, which reduced becomes 



r b r y r i> nd-y 

aj I ydydx = / / y 



ab 



t 



2 

y = 



[d-jrj 



1 + y/ a 



.(40) 



Ha = l,jf=±| 

a — co, ^ = 
a = 0, y == dL 

If # is known in equation (40), the ratio of the coefficients of elasticity may 
easily be found ; for, we have from (40) 

d-y\* E t 



- ( ( tzy\- -El 
~\ y )- e; 



(41) 



3d. Suppose that the deflecting force is not perpendicular to 
the axis, and E c = E t = E. 

Let 6 = the angle which P makes with the axis of the beam 

Fig. 29 ; 

P l = P cos 6 = the com- 
ponent of P in the direc- 
tion of the axis of the 
beam; 

P^ = P sin 6 = the com- 
ponent of P perpendicular 
to the axis of the beam ; 
Fig. 29. h = the distance of the 

neutral axis from the centre of gravity of the section AB, and 
K = the transverse section. 




FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 91 

The whole force of compression equals the whole force of 
extension, equations (37) and (38). 

. • . P cos 6 + -j- ff y dy dx = -yj J y dy dx 

But the ordinate to the centre of gravity is (see foot-note on 
page 88), 

,z ff^ dydx+ ff-yy d y dx ' 

h- y 

r.Peos0=?jKh 

PI 
or h = -^-^ cos 6 (42) 

cEK v ' 

If = 90°, h = as before found. 

If 6 = there is no neutral axis, for the force coincides with 
the axis of the beam. The equation will show the same result, 

if the value of c = - = -, equation (45), is substituted in the 

if > 

formula, for then p would be infinite, for c = 0, and h becomes 
infinite. 

4th. Let the law of resistance be according to Barlow's theory 
of flexure, and the deflecting forces normal to the axis of the 
beam. 

Using the same notation as before, also 

d 1 = the distance of the most remote fibre from the neutral 
axis, and 

(j) = the coefficient of longitudinal shearing stress. 

Then 4> j x dy = the resistance to shearing for tension, 

/o 
xdy = the resistance to shearing for compression, 
-y 

and, proceeding as we did to obtain equation (39), we have 
T /> pv py T pp° p° 

otJJq y d y dx + $J xd y —dJJ _ y d v dx + $J _ xd v-^) 

Examples. — Let the sections be rectangular, b = the breadth, d = the depth. 
Then Eq. (43) becomes 



92 



THE RESISTANCE OF MATERIALS. 



im + Mi = gj(rf- di)» + f (d - d,) 
Td> 






. \ di = id or, di = — 



20 



]J* 



the former only of which is admissible. 
If the value of G were less than that of T, the 

former would be used instead of the latter ha 
Eq. (43). 

If the section is a double T> a » in Fig". 30, 
with the notation as in the figure, <j> will be used 
in finding the resistance of the vertical rib, and 

d' 



?d 



according- to Article 75, 



d, 



cl- 



ot the lower 



Fig. 30. 



flange, and $ -r^of the upper flange. 



It appears from these several cases that the neutral axis 
passes near the centre of gravity in most practical cases, and it 
will be assumed that it passes through the centre unless other- 
wise stated. 



SHEARING STRESS. 93 



CHAPTER IV. 

SHEARING STRESS. 

89. — general statement. — Two kinds of shearing stress 
are recognized — longitudinal and transverse — both of which 
have been defined in Article 2. Materials nnder a variety of 
circumstances are subjected to this stress — such as, rivets in 
shears ; the rivets in riveted plates ; pins and bolts in spliced 
joints ; beams subjected to transverse strains ; bars which are 
twisted ; and, in short, all pieces which are subjected to any 
kind of distorsive stress in which all parts are not equally strain- 
ed. In the first examples above enumerated, all parts of the 
section are supposed to be equally strained, but the straining 
forces act in opposite directions. Shearing may take place in 
detail, as when p]ates or bars of iron are cut with a pair of 
shears, when only a small section is operated upon at a time ; 
or it may be so done as to bring into action the whole section at 
a time, as in the process of punching holes into metal, where 
the whole convex surface of the hole is supposed to resist uni- 
formly. 

90. — MODU1LXJS OF SHEARING STRENGTH. The modulus 

of resistance to shearing is the resistance which the material 
offers per unit of section to being forced apart when subjected 
to a shearing stress. 

This we call S. The total resistance to ultimate shearing 
has been found to vary directly as the section ; so that if 
K = the area of the section subjected to this stress the total re- 
sistance will be 

K.S. 

The value of S has been found for several substances, the 
principal of which are as follows : — 



94 the resistance of materials. 

Metals. 

S in lbs. per square 
inch. 

Fine cast steel * 92,400 

Eivet steel f . 64,000 

"Wrought iron * 50,000 

Wrought-iron plates punched £ .51,000 to 61,000 

Wrought iron hammered scrap punched § 44,000 to 52,000 

Cast iron 30,000 to 40,000 

Copper || 33,000 

Wood. 
With the fibres. 

White pine 480 

Spruce 470 

Firf. , 592 

Hemlock** 540 

Oak 780 

Locust 1,200 

Across the fibres. 

Eed pine 500 to 800 

Spruce 600 

Larch ft 970 to 1,700 

Treenails, English oak %% 3,000 to 5,000 

It will be seen from these results that the shearing strength 
of wrought iron is about the same as its tenacity ; of cast steel 
it is a little less than its tenacity ; of cast iron it is double its 
tenacity, and about f its crushing resistance ; and of copper it 
is about f its tenacity. 

The following table, which gives the results of some experi- 
ments upon punching plate iron, also illustrates the law of re- 
sistance, and gives the value of S for that material. 

* "Weisbach Mech. and Eng., vol. i., p. 407. 

f Kirkaldy's Exp. Inq., p. 71. 

% Proc. Inst. Mech. Eng. England, 1858, p. 76. 

§ Proc. Inst. Mech. Eng. England, 1858, p. 73. 

| Stoney on Strains, vol. ii., p. 284. 
TT Barlow on the Strength of Materials, p. 24. 
** Engineering Statics, Gillespie, p. 33. 
f f TredgoWs Carpentry, p. 42. 
X% Murray on Shipbuilding Wood and Iron, p, 94. 



SHEARING STRESS. 



95 



TABLE 

Of Experiments on Pumching Plate Iron* 



Diameter of the 


Thickness of the 


Sectional area cut 


Total perssure on 


Pressure per square 


hole. 


plate. 


through. 


the punch. 


inch of area. 


Inch. 


Inch. 


Square inch. 


Tons. 


Tons. (Gross.) 


0.259 


0.437 


0.344 


8.384 


24.4 


0.500 


0.625 


0.982 


26.678 


27.2 


0.750 


0.625 


1.472 


34.768 


23.6 


0.875 


0.875 


2.405 


55.500 


23.1 


1.000 


1.000 


3.142 


77-170 


24.6 



These results give for the value of S from 51,000 lbs. to 
61,000 lbs. The total resistance varies nearly as the cylindrical 
surface of the hole. 

APPLICATIONS. 

91 „ — problem of a tie-beam. — To find the relation be- 
tween the distance AB, Fig. 31, and the depth of a rectangu- 
lar beam below the notch, so that the total shearing strength 
shall equal the total tenacity, 




Fig. 31. 

Let A = AB = the distance of the bottom of the notch from 
the end, 
d = the remaining depth of the beam, 
h = the section of AB, 
K = the section below A, 
T — the modulus of tenacity, and 
S = the modulus of shearing strength : 



Proceedings Inst. Mech. Eng. , 1858, p. 76. 



06 



THE RESISTANCE OF MATERIALS. 



Then the condition requires that 

TK= Sk,butk:X:: hid 



Example. — For Oak 



Tc h 
'' K~ d~ 


T 


Td 




T 12000 


- 15 



15^ nearly ; hence AB should be about 



8 ~ 780 
15i times the remaining depth. 

0S2. — riveted plates. — Given the diameter of the rivets / 
it is required to find the distance oetween them from centre to 
centre^ so that the strength of the rivets for a single row shall 
equal the strength of the remaining iron in the plates. 
Let d — the diameter of the rivets, 

c — the distance between them from centre to centre, 
h = the section of the rivet, 
K = the remaining section of the plate, and 
t = the thickness of the plate. 
For iron 2 7 = S ; hence, proceeding as above, we have 



h 



~d» 



c = 



0.7S54^ 2 



K t(c-d) t 

Examples. — If t = i inch, and d = -£ inch ; 
then c = 1.2854, inch, 



+ d. 



and 



= 0.61. 



If t — % inch, and d = f inch; then c = 0.8168 and 



= 0.541, which is 



nearly the value given by Fairbairn for the strength of single riveted plates. 
See Article 27. To insure this strength the rivet should fit tightly in the hole. 

93.— TRANSVEKSE SHEARING IN BENT BEAMS. III FigS. 

24 and 25 we considered only the elongations and compressions 
of the fibres, but in transmitting the strains from the middle to 
the supports there may be a vertical force at every vertical section, 



L f A 



x 



p > z p 3 p 4 

Mil 



-1 



Fig. 32 a. Fig. 32 b. 

as is indicated by Fig. 2G, and which may be shown in a gen- 
eral way from the equations of statics. 



SHEARING STRESS. 97 

In order to simplify the problem, suppose that all the bend- 
ing forces are in a plane, and let 
P, P„ P„ &c., be the bending forces, 

F, F[, F„ &c, be the internal forces in any section of a beam, 
each of which is the resultant of all the for- 
ces concurring at that point. 
«, a x , « 2 , &a, the angles which P, P„ &c, make with the axis 

of x, 
a, a 19 a„ &c, the angles which F, F^ F„ &c, make with the 
axis of x, and y an axis perpendicular to x. 
Then the principles of Statics give the following equations : 

2P cos « + 2 Fcos a = "1 

2P sin a + 2 Fsin a = I (44) 

2{Py cos a—Px sin a) + 2 (Fy cos a — Fx sin a) = 0.] 

Let x coincide with the axis of the beam, and let all the forces 
be vertical ; or « = 90° or 270° ; then 

(1) 2Fcos a - ) 

(2) s P -f 2Fsin a = V (Ua) 

(3) s Px + 2Fy cos a — 2Fx sin a = ) 

The first of these equations shows that the sum of the resist- 
ing forces parallel to the axis is zero ; or that the total compres- 
sion equals the total tension. This is equation (35) in another 
form. The second shows that the sum of the bending forces 
equals the sum of the vertical components of the resisting forces. 
If we let 8 S represent the total strain, this equation becomes 
2P = 2F sin a =S„ which is the result sought. 

That is, when the bending forces of a beam act vertically to 
the axis of the beam, the algebraic sum of all the bending forces 
between one end and the section considered equals the vertical 
shearing stress in that section. 

The following are some of the more simple cases : 

1. Beam fixed at one end and loaded with a weight P at the 
free end, Fig. 36 8 8 =P 

2. Beam fixed at one end and loaded uniformly, Fig. 38, 
(load being w per unit of length). fi s = wx 

3. Beam supported at its ends and loaded with a weight P at 
the middle, Fig. 40 8 a ==^ P 

4. Beam supported at its ends and uniformly loaded, Fig. 42, 

S 8 = %wl —wx. 



98 THE RESISTANCE OF MATERIALS. 

5. If a beam is supported at its ends, and loaded with several 
weights P„ P 2 , P 3 , etc., as in Fig. 32 b, we have for the shear- 
ing stress. 

between the end and P, = V ; 

between P x and P 2 = V — P, ; 

between P 2 and P, = F - P x - P a ; 

between P, and P 4 = F - P x __ P 2 _ P 8 ; etc. 

If the weights are equal to each other = P, we have P = P 
— P 2 = P 3 , etc. ; and if there are n of them, and they are sym- 
metrically placed in reference to the centre of the beam, we have 

V = inP. 

If n is even, we have, at the centre of the beam, the 
transverse shearing stress = \nP — \nP = ; 
and if n is odd, there will be a weight at the centre, and each 
side of the central weight we have 
transverse shearing stress — \nP — \ (n ± 1) P = ± \P- 

These values are evidently independent of the form or mag- 
nitude of the beam. The consideration of the latter enters 
when we wish to proportion the beam to resist the former. 

The development of the third equation gives 

P x x x + P 2 x 2 + dec, -f- F x y x cos a x + F 2 y^ cos a 2 + &c. 

— F x x' sin a x — F a x" sin a a + &c = 0. 

Since x x , a? 2 , c&c, x\ x' 1 ', c&c, are linear quantities, the differ- 
ential of x x equals the differential of x 2 ; hence we have 

dx x = dx % — dx % = &c ..... = dx' = dx" = &c. 
Similarly 

^i = ^q = &c - 
Hence, by differentiating the above equation, we have 
2P dx — zFsin a dx + 2F cos a dy = 0. 

or -^ zFcos a — zFsin a — 2P. 
dx 

But the first of Eqs. (44a) reduces this equation to zero. . 
.-. sP = zFsina — S 3 . 

That is, the vertical shearing stress in a beam when the ap- 
plied forces are vertical is equal to the first differential coeffi- 
cient of the moment of the applied forces. 



SHEARING STRESS. 99 

For example, when a beam is fixed at both ends and loaded 
uniformly, the moment of the applied forces is 

as given in Eq. (122). Hence, according to the above rule we 
have 

S s = -§■ wl — wx. 
When the bending moment has an algebraic maximum, the 
moment is greatest where the shearing stress is zero ; for the 
first differential coefficient of the moment of applied forces is 
the value of the shearing stress, and this placed equal to zero 
and solved for x will give the point of greatest stress. 

The sum of the moments may be represented by a resultant 
moment. 

Let PW = 2 Px; x" 2 Fsin a = 2 Fx sin a ; and F x y' = 
2 Fy cos a. Then the third of Eqs. (44a) becomes 
P' x' - x" 2 Fsin a = F' y f 
or, P' x' - x" 8 S = F' y' 
or, P' {x' - x") = F' y' 

hence the shearing stress forms a couple with the applied force, 
or resultant of applied forces. 

If the origin of moments be taken in the section considered, 
x" will be zero, and we have 

P'x' = F y'. 
or more generally 

ZPx = SPy 

which is the fundamental equation for flexure and rupture of 
beams under transverse strains. 

94. — BENDING DUE TO TRANSVERSE SHEARING. In 

order to determine the amount of deflection due to the loading 
and transverse elasticity, it is necessary to know the law of the 
distribution of the shearing strain over the cross section. When 
the body is sheared off without deflection, as in the case of 
rivets, and other cases where the shearing force acts on the 
plane of resistance, the stress is uniformly distributed over the 
cross section; but this is not necessarily the case when the 
the shearing stress is accompanied by flexure. 

• LcfC. 



100 THE RESISTANCE OF MATERIALS. 

It will be shown in Chapter IX. that the shearing stress is 
zero at the upper and lower surfaces, and increases from- these 
points towards the neutral axis, at which point it is a maximum. 

It will be found in rectangular beams that the decrease of 
the shearing stress from the neutral axis varies as the square of 
the ordinate — Equation (210) — and hence the shearing stress 
may be represented by the area of a common parabola, the 
diminution being represented by the external part of the same 
parabola. 

Hence, if 

h = the breadth of a beam, 

d = the depth, 

A = the area of a section == fid, 

I = the length, 
E a = the coefficient of transverse elasticity: 
P = the applied weight, and 
A s = the deflection ; we have 

%E s bd = the total resistance to transverse shearing. 

The deflection will evidently increase directly as the length , 
hence, if the beam be fixed at one end and the weight be ap- 
plied at the free end 

A -. Pl 
s ~ 2 UEg 

If the beam be supported at its ends and loaded at the 
middle, we have 

_ \P.\l _,Pl 
s ~%ldE s 8 AE S 

If the beam be supported at its ends and uniformly loaded, 
we have 

. f \j wl - wx) dx = 3wl* 3 Wl_ 

As ~ """ fJA E s 16.1 Eg ISA Eg 

The total deflection depends upon the elongation and com- 
pression of the elements, as well as upon transverse shearing, 
and hence involves both E and E s . By comparing the values 
of the deflections above given, with those of the corresponding 



SHEARING STRESS. 101 

cases in the next chapter, it will be seen that the deflection due 
to transverse shearing has but little relative effect for long 
beams, bnt for very short ones it becomes the more important 
element. 

Example. — Required the deflection of a rectangular beam due to trans- 
verse shearing which is supported at its ends and loaded uniformly over its 
whole length, when b = 4 inches, d = \Q inches, 1=8 feet ; w (the load per 
foot of length) = 500 pounds, and E s = 45,000 pounds. 

Weisbach says: "The coefficient of transverse elasticity is 
assumed to be J E " (Weisbach, Mechanics of Engineering, 
Yol. I., p. 522). This is nearly f E, the value found theoreti- 
cally for amorphous bodies, but for fibrous bodies, such as wood, 
the transverse elasticity is not the same in the different directions 
of the layers, so that it has not a specific value. 

There is a longitudinal shearing at every point of a beam 
where there is transverse shearing, but the deflection which 
arises from it is small. The analysis of these cases is reserved 
for Chapter IX., since a portion of it depends upon the analysis 
for flexure. 



95. shearing- resistance to torsion.— "When a solid 
is twisted the consecutive transverse sections of elements slip 
over each other, and for small angles of torsion, such as are 
only admitted in practice, the law of strains is comparatively 
simple, as is shown from theoretical considerations, and which 
is confirmed by experiment. This law is : the strains increase 
directly as the distance from the axis of the piece, as stated in 
Chapter VIIL, and is applicable to wood and other fibrous and 
granular solids. 

But when the elastic limit is passed the case becomes very 
complex. All the elements which originally were rectilinear 
become helices, except those at the axis. The outer elements 
thus become elongated, and by their elastic resistance produce 
compression upon those near the axis. There will alsr be a 
lateral contraction of those elements which are elongated. The 
transverse sections which were originally plane will become 
warped. As the strain is increased the outer elements actually 



102 THE RESISTANCE OF MATERIALS. 

slip over each other, and thus lose, in a great degree, theii 
power of resistance, and throw greater strains upon those nearer 
the axis, until finally the elements are sheared apart. During 
this process, shearing strains may exist in any direction — longi- 
tudinally, lateral, and tangential. 

In the more ductile metals rupture may take place slowly, 
and the final fracture be nearly a plane which will be perpen- 
dicular to the axis of the piece ; but in brittle metals, such as 
highly tempered steel and most qualities of unmalleableized 
cast-iron, rupture takes place suddenly with a " snap " when 
under strain, and with only a small amount of torsion. In 
such cases the fracture is irregular and oblique to the axis. 
There is little or no appearance of shearing, for rupture takes 
place with only a small amount of shearing. 

The conditions of ultimate rupture do not appear to be gov- 
erned by definite mathematical laws ; and hence it might ap- 
pear useless to subject them to hypothetical laws ; but the laws 
which are assumed are sufficiently exact for practical cases when 
the material is not overstrained. 

Remark. — It is fortunate that for practical purposes it is not necessary to 
know the exact condition of the strains within a piece which is used in a 
structure, for it is impossible to construct an equation which will represent 
every possible case with mathematical exactness. Bodies are infinitely diver- 
sified. Some may be subjected to internal strains from the process of manu- 
facture. These may be caused by f orging some parts more than others ; but 
especially bj unequal cooling. The effect of an external load may be to in- 
crease the intensity of some of these strains and relieve others. We also see 
that a simple stress may produce various strains : and hence when the bodies 
are free from internal strains, and are perfectly homogeneous, the analysis 
which considers all the changes becomes exceedingly refined. 

We know by long experience that it is only necessary to keep within certain 
limits, and these limits can easily be determined. 

In the analysis of the more simple cases we consider only one distortion at 
a time. Thus, in 'stretching a piece, we consider the more apparent phenom- 
enon — that of elongation — but at the same time there is a lateral contraction 
which, in practice, is so small that we disregard it, but which in a thorough 
analysis must be considered. Also in regard to flexure, we usually consider 
only the effect of the elongation of some of the fibres and the compression of 
others, as in the following chapter, but this change is necessarily accompanied 
by others, which in ordinary cases may be disregarded. The same remarks 
apply to torsion and to transverse shearing. 

The analysis which determines the relation between strains and stresses in 
elastic bodies has given rise to a department of mathematical physics called 



SHEARING STRESS. 103 

the Mathematical Theory of Elasticity, which has heen developed by M. Lame 
{Lemons de la Theorie MatMmatique de V Elasticity des Corps, Solides, Paris, 1852) ; 
M. Louiville (Journal Louiville, 1863, etc.); M. Kirchoff (TJeber das Qleich- 
gewicht und Bewegung einer unendlich dunnen elastiches Stabes* Journal de 
Crelle, tome 56, p. 285); M. Maxwell (On the Equilibrium of Elastic Bodies; 
Transactions of the Royal Society of Edinburgh, vol. xx., 1853, p. 87, etc.)' 
M. Cauchy (Exercises d 'analysis et de P7iysique Mathematique ; Comptes Bendus, 
etc.), and others. 

* On the equilibrium and movement of an infinitely slender elastic rod. 



104 



THE RESISTANCE OF MATERIALS. 



CHAPTER V. 




Fig. 33. 



FLEXURE. 

96. ELASTIC CURVE. 

When -a beam is bent by a transverse strain, equilibrium is 
established between the external and internal forces ; or, to be 
more specific, all the external forces to the right or left of any 

transverse section are held 
T in equilibrium by the elastic 

resistances of the material 
in the section. When in 
this state the curve assumed 
by the neutral axis is called 
the elastic curve. 

We will first find the 
equation of the elastic curve 
according to the conditions 
of the " Common Theory," following substantially the method 
originally given by Kavier. Let Fig. 33 represent a beam, fixed 
at one end, or supported in any manner, and deflected by a 
weight, P, or by any number of forces. AB is the neutral 
axis. Take the origin of co-ordinates at B (or at any other 
point on the neutral axis), and let x be horizontal and coincide 
with the axis of the beam before flexure, y vertical and u per- 
pendicular to the plane of xy. The transverse sections CM 
and EF being consecutive and parallel before flexure, will 
meet after flexure, if sufficiently prolonged in some point, as o. 
Through iV~ draw KH parallel to CM; then will fce be the 
elongation of a fibre whose original length was ck. We have 
the following notation : — 

dx = LN — the distance between consecutive sections, 
y f = JVe = any ordinate of the surface, 
u = JVa or JVa'. 

b = NN' = the limiting value of u, 
f {y\v) = equation of the transverse section, 



FLEXUEE. 105 

dy'du = the transverse section of a fibre, 

f (x, y) = the equation of the neutral axis, 

p = ON = the radius of curvature at J¥, 

p = the force necessary to elongate any fibre an amount 
equal to a when applied in the direction of its length, 

a = Jce, 

I == the moment of inertia of the section, 

E = the coefficient of elasticity of the material, which is 
supposed to be the same for extension and compression, 

s Px = a general expression for the moment of applied 
forces. 

We suppose that the strain is within the elastic limit, and 
establish the algebraic equation on the condition that the sum 
of the moments of the applied or deflecting forces equals the 
sum of the moments of the resisting forces. We also assume 
that the neutral axis coincides with the centre of the transverse 
sections of the beam. 

By the similarity of the triangles Z^i^and kffle, we have 
OJST: Ne \ \ LN : ke, or p : y' \ \ dx : a 

.-.a — ILdx (45) 

P 

The force necessary to produce this elongation is (see Equa- 
tion (3) ), 

p = E dy'du A ; 

which becomes, by substituting a from (45), 

p = -y' dy'du (46) 

P 

and the moment of this force is found by multiplying it by y f ; 
jp 

. • . py' — —y n dy'du (47) 

P 
The total moment of all the resisting forces to extension and 
compression is found by integrating Equation (47) so as to in- 
clude the whole transverse section, and this will equal the sum 
of the moments of the applied forces : 



U J/ dy ' du+ f J4 wdw l 



2P 



106 THE RESISTANCE OF MATERIALS. 

or- / j y , *dy / du = 2Px (48) 

The quantity E f fy' 2 dy'du, which depends npon the form of 

the transverse section and nature of the material, is called the 
moment of flexure. 

The quantity f fy n dy'du, when taken between limits so as 

to include the whole transverse section, is called the moment of 
inertia of the surface.* Calling this / and Equation (48) 
becomes 

y =*P* (49) 

which is the equation of the elastic curve. 

An exact solution of Equation (49) is not easily obtained in 
practice, except in a few very simple cases ; but when the de- 
flection is small an approximate solution, which is generally 
comparatively simple and always sufficiently exact, is easily 
found. 



(i + ^V 



dx* 



d 2 y dx o?y 

dx* 
— — - nearly, since for small deflections 
d*y 

•— (which is the tangent of the angle which the tangent line to 

the curve makes with the axis of x) is small compared with 
unity, and hence may be omitted. Hence equation (49) becomes 

EI^^ZPx ....(50) 

which is the general approximate Equation of the neutral axis. 

97. the moment of inertia of all transverse sections 
of a prismatic beam is constant, and hence I is constant for 
prismatic beams. 

* See Appendix. 



FLEXURE. 



107 



For a rectangle, as Fig. 34, we have 
+ id 

y*dydu = ^bd? (51) - 



// 



For a circle, the origin of coordinates being at 
the centre ; 

y = p sin 
dydu = p dp dd 



Jo JO 



2tt 



p a dp dd sin 3 = far r 4 



Fig. 34. 




Fig. 35. 



(52) 



special cases of prismatic beams. 

08. Required the equation of the neutral axis, amount 
of deflection, and slope of the curve of a prismatic beam, 
when slightly deflected, and subjected to certain conditions 
AS follows : 

09. case i. — Suppose a horizontal beam is fixed at one 

EXTREMITY AND A WEIGHT P RESTS UPON THE FREE EXTREMITY ; 
REQUIRED THE EQUATION OF THE NEUTRAL AXIS AND THE TOTAL 
DEFLECTION. 




Fig. 36. 




Fig. 37. 



The beam may be fixed by being embedded firmly in a wall, 
as in Fig. 36, or by resting on a fulcrum and having a weight 
applied on the extended part, which is just sufficient to make 



108 THE RESISTANCE OF MATERIALS. 

the curve horizontal over the support, as in Fig. 37. The latter 
case more nearly realizes the mathematical condition of fixed- 
ness. In either case let 

I = AB = the length of the part considered, 
i = the inclination of the curve at any point, and 
A = BO = the total deflection. 

Take the origin of coordinates at the free end, A / x horizontal, 
y vertical and positive downwards. The moment of P on any 
section distant x from A is Px, which is the second member of 
Equation (50) in this case. Hence Eq. (50) becomes 

UI%=P* (53) 

Multiply both members by dx and integrate, and we have 

£7^ = iP*?+C 1 (54) 

When the deflections are small, the length of the beam re- 
mains sensibly constant, hence for the point B,x = l\ and at 

on 
the fixed end -~ = 0. Substitute these values in Eq. (54), and 

we find C\ — — J PP, and (54) gives 
The integral of Equation (55) is 
But the problem gives y = for x = .*. C % = ; 

■■■y = Si^- spx) (56) 

which is the equation of the neutral axis, according to the com- 
mon theory, and may be discussed like any other algebraic 
curve. 

The greatest slope is at A, to find which make x = in Equa- 
tion (55) 

PP 
.*. tang i (at the free end) == — kt^t 



FLEXUKE. 



109 



ax 6 



The greatest distance between the curve and the axis of x is 
at B, to find which make x — I in Equation (56), and we have 

y = A = -rm w 

In this case we have 

*£»-f ...........■<«., 

That is, the transverse shearing strain is uniform over the 
whole length and equal to the load at the free end. 
Differentiating again gives 

— — — o # 

that is, the increment of transverse shearing is zero. 

If y were positive upward, everything else remaining the 
same, the second member of Eq. (53) would have been negative, 
for it is a principle in the Differential Calculus that when the 
curve is concave to the axis of x, the second differential coeffi- 
cient and the ordinate must have contrary signs. This would 
make tang i and A positive. It will be a good exercise for the 
student to solve this and other problems by taking the origin 
of coordinates at different points, only keeping x horizontal and 
y vertical. For instance, take the origin at B ; at O; at the 
point where the free end of the beam was before deflection ; at 
the middle of the beam ; or at any other point. 

Example.— If I = 5 ft., b = 3 in., d= 8 in., E= 1,600,000 lbs., and P= 5,000 
lbs. ; required the slope at the free end and at the middle, and the maximum 
deflection. 

£©©. case ii. — Suppose that the beam is fixed at one 

END, IS FREE AT THE OTHER, AND HAS A LOAD UNIFORMLY DISTRI- 
BUTED over its whole length. — The beam may be fixed as 
before, as shown in Figs. 38 and 39. 




Fig 




110 THE RESISTANCE OF MATERIALS. 

Let w = the load on a unit of length. This load may be the 
weight of the beam, or it may be an additional load. 
W= wl = the total load. 
Take the origin at A* 

Then wx = the load on a distance x, and 

fywx 2 = the moment of this load on a section distant 
x from A. 
Hence Equation (50) becomes 

Fl^^iwa?. (58) 

.-. J = eJj^- *)-*«. (59) 

:.y = ^ m {^-4:Fx) (60) 

a a wli WI ' nm 

In which -^ = f or x = I .*. C x = — n „ r > 
ax 6A1 

y = for x = .*. C 2 = 0, and 
y = A for x — I. 

If the origin of coordinates were at the fixed end, 2 Px in the 

w 
first case would be P (I — x), and in the second -~ (I — a?) 2 . The 

student may reduce these cases and find the constants of inte- 
gration. This case may be further modified for practice by 
taking the origin of coordinates at different points. 
From Eq. (58) we have 



Also 



Ss = EI^ = wx . (62a) 

d Ss = EI -=r-~ — wdx\ 
dx* ' 



that is, the increment of shearing is the load per unit of length 
multiplied by the increment of length. 

101. case in. — Let the beam be fixed at one end and a 

LOAD TTNDTORMLY DISTRIBUTED OVER ITS WHOLE LENGTH, AND A 



FLEXUKE. 



Ill 



weight also applied at the feee end. — This is a combination 
of the two preceding cases, and is represented by Figs. 36 and 
37, in which the weight of the beam is the uniform load. 



Z 3 



,(62) 



and A = -^{P + iW) 

hence the deflection of a beam fixed at one end and free at the 
other, and uniformly loaded, is f as much as for the same weight 
applied at the free end. 

10&. case iv. — Let the beam be supported at its ends 
and a weight APPLIED at ant point. — Figs. 40 and 41 repre- 
sent the case. 





Fig. 41. 

Let the reaction of the supports be V and V v Take the 
origin at A over the support, and let AD = c = the abscissa 
of the point of application of P. 

Then, V= ^ P, and V x = j P. 

The case is the same as if a beam rested on a support at Z>, 
and weights equal to V and V x were suspended at the ends. 
For the part AD, Equation (50) becomes : — 






' dx 



Vx = -~Px 






and y = ~ F tiEi > x * + 0ix+( - ' = °)- 



.(63) 
•(64) 
.(65) 



in the last of which, y = for x = .*. C % = as indicated. 



112 THE RESISTANCE OF MATERIALS. 

For the part DI>, the origin of coordinates remaining at A- 
we have : — 

EI % = - v *+ F (°-<o = FcX -j~ l = - ^ ('-*)• • • ( 66 ) 
^%=&h^-^ +G ' < 67 > 

and y = ^- J (x s -Slx i ) + Cx + C" (68) 

To find the constants, make x = c in equations (64) and (67) 
and place them equal to each other ; do the same with (65) and 
(68) ; and also observe that in (68) y = for x — I. These 
conditions establish the three following equations : — 

-^8? )+ ^S(<<- 2 ^' 

From these we find 

Pr 

° - 6£I 
Hence, for the part AD we have 
„ r d?y l — o-r, 

SJ ~ *wi af + e£n^ + M M > 

0r ' % = e^("(-«+8fl)rf+^ + 2oP-8<«l (69) 

y= -^jX{o-iyH<? + ^-^T)c^\ (70) 



FLEXTJKE. 113 

To find the maximum deflection, if c is greater than JZ, make 

-r- = '0in (69) and find x ; then substitute the value thus found 

(Jij 
in Equation (70). If c < \l make-y = in Equation (67) and 

substitute the value thus found in Equation (68). 

If D is at the middle of the length, make c = \l in equations 
(63), (69), and (70) ; and we have for the curve AD 

EI^=-\Px (71) 

dx~ = TemP" 4 **)' 

y=~{^-^) • (72) 

and A = ^gj-Of x = ¥in (73) >. (73) 

The greatest stress is at the centre, and the maximum mo- 
ment is found by making x = -p in the second member of 
Equation (71). Hence, the maximum moment is 

\Pl (73a) 

In this case the curve DB is of the same form as AD, but 
its equation will not be the same unless the origin of coordi- 
nates be taken at the other extremity of the beam. 

From Eq. (71) we have 

MsodSs = 0. 

103. case v. — Suppose that a beam is suppoeted at ok 

NEAR ITS EXTREMITIES, AND THAT A LOAD IS UNIFORMLY DISTRIB- 
UTED OVER ITS WHOLE LENGTH. 

No account is made of the small por- . 
tion of the beam (if any) which projects 
beyond the supports. The distance be- ■! 
tween the supports is the length of the 
beam which is considered. %^ 

Let the notation be the same as in the fig. 42 

preceding cases. 
8 




114 THE RESISTANCE OF MATERIALS. 

Then V — \wl = \ W= the weight sustained by each support; 

Vx = iwlx == the moment of V on any section, as c ; 

wx is the load on x, and the lever arm of this load is 
the horizontal distance from its centre to the section <?, or %x ; 
hence its moment is ^tox 2 , and the total moment is the differ- 
ence of the two moments. Hence Equation (50) becomes 

jgT^=4w(-fc+-«f) (74) 

andif« = Vm(75),y = J = - S gj Z? = S g izr (76) 

In these equations -p = for a? = $1, /. 6i = oZ^r> 

and y = for x = 0, ,\ {7 2 = 0. 
$s = -JwZ — wa?. 



104. case vi. — Let the beam be supported at its ends, 

UNIFORMLY LOADED, AND ALSO A LOAD MIDWAY BETWEEN THE 
SUPPORTS. 

This case is a combination of the two preceding ones, and 
may be represented by Fig. 40 ; for the weight of the beam 
may be the uniform load. Hence 

EI M = " iFx + *™^ "" ^° lx (77) 



!>+§<] m 



4&JZIIT ' 8 

Experiments on the deflection of beams are generally made 
in accordance with this case. If the beam be rectangular, we 
have from Equation (51), 



FLEXURE. 115 

I = — bd 3 , which in (78) gives 

J =imd p+ -s w 1 w 

•■■ E =md. p +l w l<- ( 80 > 

According to Equation (79) the deflection of rectangular 
beams varies as the cube of the length ; and inversely as the 
breadth and cube of the depth, and directly as the weight 
applied. 

In making an experiment to determine E, the beam is 
weighed, and that portion of it which is between the supports 
and unbalanced will be W, and all the quantities except E 
may be directly measured. If E be known, we may measure 
or assume all but one of the remaining quantities, and solve 
the equation to find the remaining quantity, as the following 
examples will illustrate : — 

Examples. — 1. If a rectangular beam, 5 feet long, 3 inches wide, and 3 inches 
deep, is deflected -£$ of an inch by a weight of 3,000 lbs. applied at the middle ; 
required the coefficient of elasticity. E = 20,000,000 lbs. 

2. If b = 2 inches, d = 4 inches, and I = 6 feet, the weight of the beam 
144 lbs., and a weight P— 10,000 lbs. placed at the middle of the beam deflects 
it i an inch ; required E. E = 14,711,220 lbs. 

3. A joist, whose length is 16 feet, breadth 2 inches, depth 12 inches, and 
coefficient of elasticity 1,600,000 lbs., is deflected $ inch by a weight in the 
middle ; required the weight ; the weight of the beam being neglected. 

Ans. P = 1,562 lbs. 

4. An iron rectangular beam, whose length is 12 feet, breadth H inch, co- 
efficient of elasticity 24,000,000 lbs., has a weight of 10,000 lbs. suspended at 
the middle ; required its depth that the deflection may be 4^0 of its length. 

Ans. 8.8 in. 

5. A rectangular wooden beam, 6 inches wide and 30 feet long, is supported 
at its ends. The coefficient of elasticity is 1,800,000 lbs. ; the weight of a 
cubic foot of the beam is 50 lbs. ; required the depth that it may deflect 1 
inch from its own weight. 

How deep must it be to deflect -j-foj of its length ? 

6. A cylindrical beam, whose diameter is 2 inches, length 5 feet, weight of 
a cubic inch of the material 0.25 lb., is deflected | of an inch by a weight 
P = 3,000 lbs. suspended at the middle of the beam. Required the coefficient 
of elasticity. 



116 THE RESISTANCE OF MATERIALS. 

To solve this substitute I— jTrr 4 (Equation (52)) in Equation (78). This gives 

7. Required the depth of a rectangular beam which is supported at its ends, 
and so loaded at the middle that the elongation of the lowest fibre shall equal 
T4V0 of its original length. (G-ood iron may safely be elongated this amount.) 

ET AET 

Equations (49) and (73a) become — =iPL'.p= -^-. In this substitute the 

p Jrli 

value of I, Equation (51), and it becomes 

p = Md \ By the problem find p = 700tf 

7 2100P* 



,.<^ 



Eb 



8. Required the radius of curvature at the middle point of a wooden beam, 
when P= 3,000 lbs. ; Z = 10 ft. j b = 4 in. ; d = 8 in. ; and E = 1,000,000 lbs. 

-ci « „ m * ,*« v . EI 1,000,000 x-fVx 4 x8 3 , ontf 

Equates (49) and (78a) give p = —= ^ m ^ 1Qxl2 = 1,896 

inches. 

9. Let the beam be iron, supported at its ends. Let b = 1 in., d = 2 in., 
I = 8 ft., E— 25,000,000 lbs. Required the radius of curvature at the middld 
when the deflection is \ of an inch. Use Eqs. (49) and (73) for P at the middle. 

72 
z-r-. = 3,840 inches ; 



EI 




E.I 


ipr 


"1 


4SE.I.J.I 




P 



from which it appears that it is independent of the breadth and depth. 

10. The centrifugal force caused by a load moving over a deflected beam 



may be found from the expression , in which m is the mass of the moving 

load, v its velocity in feet per second, and p the radius of curvature of the 
beam. (See Mechanics.) 

11. All these problems may be applied to beams fixed at one end, and P 
applied at the free end, or for a load uniformly distributed over the whole 
length, by using the equations under Cases I. , II. , and III. 



105. case vii. — Let the beam be fixed at one extrem- 
ity, SUPPORTED AT THE OTHER, AND HAVE A WEIGHT, jP, APPLIED 
AT ANY POINT. 

The beam may be fixed by being encased in a wall, Fig. 43, 
or by extending it over a support and suspending a weight on 



FLEXURE. 



117 



the extended part sufficient to make the beam horizontal over 
the support, Fig. 44 ; or by resting a beam whose length is 21 





Fig. 44. 



on three equidistant supports, and having two weights, each 
equal to D, resting upon it at equal distances from the central 




Fig. 45. 

support, Fig. 45. In the latter case each half of the beam 
fulfils the condition of the case. 

Let I — AB, Fig. 43, be the part considered, 
V= the reaction of the support, 
nl — AD — the abscissa of P, and 
f = the deflection of the beam at D. 

Take the origin at A, the fixed end. We may consider that 
the curve DB is caused by the reaction of V, while all the 
forces at the left of P hold the beam for V io produce its 
effect. Similarly the curve AD is produced by the reaction V 
and the weight JP, while all the forces at the left of them hold 
the beam. In all cases we may consider that the applied forces 
on one side of the transverse section are in equilibrium with 
the resisting forces of tension and compression in the section. 
It is well also to observe that the origin of moments is at the 
centre of the transverse section, while the origin of coordinates 
may be at any point. 

For the curve AD we have, observing that -Jf- = f or x = 0. 
and y = for x = : — 



118 THE RESISTANCE OF MATERIALS. 

EI^=P{nl-x)-V{l-x) ,....(82) 

J^-'O*-*)-^-*) < 83 > 

****(¥-i)-T<g-i) w 

For the point J), we have, by making x — nl, 
|£ = tan ^ftrfP -(»-*»») 7] ^- (85) 

y =/= [> 8 P-(£» S ~K) F] ^ (86) 

For the curve i>^, observe that -r- = tang i for a? = wZ, and 

y =yfor a; = nl, nsing for their values (85) and (86) in deter- 
mining the constants in the following equations, and we have : — 

EI %=- v y-^ < 87 > 

EI%=*Pt*fi-Vq*-^ (88) 

Ely = (*a - *»J)Pn»P - r(~-j) (89) 

To find the reaction V, observe that y = 0, f or x = Z in (89), 
and we obtain : — 

... V= in\S - w)P (90) 

By substituting this value of Tin the preceding equations, 
they become completely determined. For the curve AD we 
have : — 

£Ijj! = P{nl - x- W(3 -»)(*-*)] ,(»1) 

2 = ^[4»&-2a?-^(3-»)(2&-a?)] (92) 



FLEXURE. 119 

y = ^j \§nh* - 2a? - n\Z - n) (SW-a?)] . . . .(93) 
and for the curve DB : — 

EI^ = -hPn\Z-n){l-x) (94) 

iha£p*-fl-">fl i "-«'W (95> 

Pn 2 

y = j~j [ (6x - 2n£p - (Six 2 - x*) (3-n) ]. . . .(96) 

The points of greatest strain in these curves are where the 
6um of the moments of applied forces is greatest, and this is 
greatest when the second members of (91) and (94) are greatest. 
Neither of these expressions have an algebraic maximum, and 
hence we must find by inspection that value qfx which will 
give the greatest value of the function within the limits of the 
problem. Equation (91) has two such values, one for x = 0, 
the other for x = nl, and Equation (94) has one such for x = nl, 
which value will reduce (91) and (94) to the same value. 

Making x = in (91) gives for the moment of maximum 
strain, 

SPx = iPl[2n-3n 2 + n 5 -].. (97) 

For the moment of strain at P, make x = nl, in Eq. (91) or Eq. 
(94), and we have 

SPx = i Pin 2 [- 3 + ±n - n 2 ] (98) 

To find where P must be applied so that the strain at the point 
of application shall be greater than if applied at any other 
point, we must find the maximum of (98) : — 

...^^.= = - 6» + 12ri> -M (99) 

dn ; 

n = 0.634 + (100) 

or the force must be applied at more than T 6 ^- of the length of 
the beam from the fixed end. This value of n in (98) gives, 

2Px = Plx 0.174 

Equation (99) has two values of n, but the other is not within 
the limits of the problem. 



120 THE RESISTANCE OF MATERIALS. 

The position of the weight, which will give a maximum 
strain at the fixed end, is found by making (97) a maximum. 
Proceeding in the usual way, we find : — 

n - 1±WS = °- 422 + ( 101 ) 

which in Eq. (97) gives, SPx = PI x 0.181 (102) 

and in Eq. (98) XPx = PI x 0.131 + . 

To find where P must be applied so that the strain at the 
point of application will equal the strain at the fixed end, make 
Equations (97) and (98) equal to each other, and find n. This 
gives, 

n = \ 3.4141 + (103) 

( 0.5858 +. 

But n = 0.5858 + is the only practical value. 

To find where P must be applied so that the curve at that 

cLij 
point shall be horizontal, make -^ = 0, and x = nl in (95). 



1. 

This gives n — \ 3.4141 

0.5858 



-{ 



which are the same as the preceding values of n. To find the 
corresponding defection, make x = nl, and w = 0.5858 +, in 
(93), and we find 

A = 0.0098 2 ~ (104) 

For n < 0,5858, tang i is 4- ) 

n > 0.5858, tang »&—>.... (105) 

n =s 0.5858, tang i is ) 

To find the maximum deflection when n — 0.634, make ~r = 

' dx 

in Eq. (92) or (95), according as the greater deflection is to the 
right or left of P. But, according to Eq. (105), it belongs to the 
curve AD ; hence use Eq. (92). Making n = 0.634 in Eq. (92^ 
placing it equal zero, and solving gives, 

a = 0.6045Z; 



FLEXtTEE. 121 

which in Eq. (93) gives, 

y = A = 0.00957-^ (106) 

To find where P must be applied so as to give an absolute 
maximum deflection; first find the abscissa of the point of 
maximum deflection, when P is applied at any point by making 

■j. — in Eq. (92), and thus find 

2(3 - n)n* - 4w 7 Mnm 

X = ->5 fl cT l ( 107 ) 

(3 — rijn* — 2 x ' 

which, substituted in Eq. (93) gives the corresponding maximum 
deflection. Then find that value of n which will make the 
expression a maximum. 

To find the deflection when P is placed at the middle, make 
n*= J in Eq. (93) or Eq. (96), which gives 

._ 7 PP 
~~ 48.16 EF 

The point of contra-flexure in the curve AD is found by 
making -^ = in (91) (see Dif . Cal.) which gives, 





3ri? — n B — 2n. 


h * = A^- 





The second member of Eq. (91) is the moment of applied 
forces, and as it is nought at the point of contra-flexure, it fol- 
lows that at that point there is no bending stress, and hence no 
elongation or compression of the fibres, but only a transverse 
shearing stress. The value of the transverse shearing is 

Ss = EI H = - P + in 2 (3 - n) P 

which compared with Eq. (90) shows that the shearing strain 
at any point of the curve AD subtracted from the reaction at 
B equals the total load P„ 

If a beam rests upon three horizontal equidistant supports, 
and two weights, each equal P y are placed upon it, one on each 



122 



THE RESISTANCE OF MATERIALS. 



side of the central support and equidistant from it, it fulfils 
the condition of a beam fixed at one end and supported at the 
other, as before stated, and the amount which each support will 
sustain for incipient flexure may easily be foun<J from the pre- 
ceding equations. 

The three supports will sustain 2P, and the end supports 
each sustain F= in\S - n)P. (See Eq. (90).) 

Hence, the central support sustains 

F = 2P - n\Z - n)P. 
If n = i,V=&P, and P = ff P. 

106. case viii. — Let the beam be fixed at one end, sup- 
ported AT THE OTHER, AND UNDJORMLY LOADED OVER ITS WHOLE 



LENGTH, 





Fig. 45. 



Fig. 46. 



Take the origin at A, Figs. 45 and 46, and the notation the 
same as in the preceding cases, then Equation (50) becomes 



EI^ = iwx* - Vx. 



(108) 



Integrating gives J = ^(rf - Z 3 ) + ^ 0" ~ «f), ( 109 ) 



w V 
and y = ^M^ ~ 4 ^) + 6^r( 3 ^ ~^) ( ll0 ) 

in which -¥• = for x = 1, and y = for a? = 0. 

If V= 0, these equations become the same as those under 
Case II. 

In Equation (110) y is also zero, for x = I ; for which values 
we haveF= | TF= %wl (Ill) 

This value substituted in Equations (108), (109), and (110) 
gives : — 



FLEXURE. 123 

ii70 = i«w(to-J«) (112) 

2=szr<*'-M* + * (113) 

on 

y = ±$£ I V* i -zw + P*) ( 114 > 

The point of maximum deflection is found by placing Equa- 
tion (113) equal zero and solving for x. This gives 

x= I, 
and, x = 1± 1 ^ 83 1 = 0.4215Z, 

using the positive value only ; and this in Eq. (114) gives 

IF/ 3 
y = A = 0.0054 -^ (115) 

There are two maxima strains ; one for x = Z ; the other for 
a? == f I. The former in (112) gives 

SPx = %wl* = iWl (116) 

and the latter gives 

SPx =-^Wl=-^Wl nearly. 

The point of contra-flexure is found from Equation (112) to 
be at x = f I, at which point the longitudinal strains are zero, 
and there is only transverse shearing. 

From Eq. (112) we have 

Ss = EI-j! = wx — fwl 

For x = $1 we have Ss = %wl (116a) 

If the oosm is supported by three props, which are in the 
same horizontal, Fig. 46, then each part is subjected to the same 
conditions as the single beam in Fig. 45. Hence, if W is the 
load on half the beam, each of the end props will sustain V = 
%W (Eq. (Ill)), and the middle prop will sustain 2 W-%W 

From the supported end, A, to the point of contra-flexure 



124 



THE RESISTANCE OF MATERIALS. 



(f I) the beam is in the same condition as a beam which is sup- 
ported at its ends and uniformly loaded. Hence the supported 
end sustains i of $wl = %wl, as before found. The shearing 
strain at the point of contra-fiexure must be the same as at the 
supported end, which agrees with Eq. (116a). 

Such are the teachings of the " Common theory." But the 
mathematical conditions here imposed are never realized. It is 
impossible to maintain the props exactly in the same horizontal. 
As they are elastic they will be compressed, and as the central 
one will be most compressed, the tendency will be to relieve the 
strain on it and throw a greater strain upon the end supports. 
If the supports be maintained in the same horizontal, the results 
above deduced will be practically true for very small deflec- 
tions, within the elastic limits. 

107. case ix. — Let the beam be fixed at both ends and 

A WEIGHT REST UPON IT AT ANY POINT. 



To simplify the case, sup- 
pose that the weight rests 
at the middle of the length. 

Let the beam be extend- 
ed over one support and a 
weight, P l rest at C, suffi- 
cient to make the curve hori- 
zontal over the support A. 
We have V= P x + \P. 

Let A O—qL 

Then for the curve AD we have, 

EI^=P x (ql + x)- Vx = P iq l-iPx 




Fig. 47. 



ax 



-iPz*+ (O L = 0). 
dy 



To find P x observe that -f- = for x — \l\ 

This reduces the preceding equations to the following 



FLEXURE. 125 

£T^=iP(l-<kc) (117) 

g^0— *> (HO) 

and by integrating again, we find : — 

y = SSJEzC8W-4rf):...' (119) 

PV 
Forx^ilm(lW),rj = A= JMWl (120) 

There is no algebraic maximum of the moment of strain as 
given in the second member of Equation (117), but inspection 
shows that within the limits of the problem the moment is 
greatest for x — or x = \l. These in (117) give the same 
value, with contrary signs; hence the moment of greatest 
strain is 

SPx = ± \Pl (121) 

The moment is zero for x — JZ. 

108. case x.— Let the beam be fixed at both ends and 

A LOAD UNTFORMLY DISTRIBUTED OVER ITS WHOLE LENGTH. 




Fig. 48. 



The notation being the same as before x sed v we have 

V= P, + iwl 

Letql = AC. 

The equation of moments is 



Integrating, and observing that -y- = f or x = ; also y = 



Z/^ = \wx* - fyolx* + Pgfa 



126 THE RESISTANCE OF MATERIALS. 

EI^ = iwx 2 -Vx + i\(# + x) 

= %wx* — Jwfe + P-fil. 

3rv 
for a? = 0, and we have 

Elf 

ax 

Ely = t^wx* - &wlx* + iPtfk?. 

civ 
But j- = for a? = ? ; also y = for a? =? ; 

1 wl W 

'"" X ~ 12 gr - 12^ 

which substituted in the previous equations give : — 

^S-bC'-^M] (122 > 

f=r5r^-^-2^] (123) 

»=h5b <*-">* •••( 12 *> 

1 TP7 3 
For< B = iZin(124),2/ = J = 3g i - Er (125) 

Making -J$ = we find for the points of contra-flexure 



IS: 



7887? 
2113? 



at which point there is no longitudinal strain, but a transverse 
shearing strain. We have 

& = -as- =ir [2a! - !1 

which is equal to-JTFat the ends (either ±) and zero at the 



FLEXURE. 127 

middle. At the first point of contra-flexure (x = 0.2113Z) the 
shearing strain is 0.2887 W, to which add the load on that part, 
= 0.2113 W, gives 0.5000 W, or \ the total load. 

The maximum moments are for x = and x = %l. 
For x = 0, the second member of Eq. (122) gives -^ Wl. (i26) 
Fora = iZ, " " " — &TP7. 

Hence the greatest strain is over the support, at which point 
it is twice as great as at the middle. If W = i 3 , we see that 
the strain over the support is f as great in this case as in the 
former. 



128 THE RESISTANCE OF MATERIALS. 

109. RESTJTLTS COLLECTED. 



No. OF 

THE 
CASE. 


Condition or 

THE BEAM. 


HOW 
LOADED. 


General 

moment of 

FLEXURE. 


Maximum 
moment of 

STRESS. 


« i 


Relative 

Max. DEFLEC- 
TION OR CO- 
EFFICIENT OF 

EI 


I. 


Fixed at 

ONE END. 


Load at 

FREE END. 


Px. 

Eq. (53). 


PI. 


24 


Eq. (57). 


II. 


Uniform 
load. 


jWX 2 

Eq. (58). 


i'Wl 


12 


Eq. (61). 


IV. 


Supported 

AT THE 

ENDS. 


At the 

MIDDLE. 


\Px. 

Eq. (71). 


IPI. 


6 


-hP- 
Eq. (73). 


V. 


Uniform. 


\w{lx-x z ). 
Eq. (74). 


iWl. 


3 


Eq. (76). 


VII. 


Fixed at 

ONE END 
AND SUP- 
PORTED AT 
THE OTHER. 


At 0.634? 

FROM 

FIXED END 

Eq. (100). 


For AD 

Eq. (91). 
ForD# 
Eq. (94). 


f(2 VS-S)PI 


4 + 


^nearly. 
Eq. (104). 


VIII. 


Uniform. 


$w(4:X 2 -Slx). 
Eq. (112). 


Eq. (116). 


3 


W 

_ nearly. 

Eq. (115). 


IX. 


Fixed at 
both ends. 


At the 
middle. 


$P(l-4x). 
Eq. (117). 


Eq. (121). 


3 


P 

192* 

Eq. (120). 


X. 


Uniform. io7 v ' 
} dL Eq. (122). 


-h-wi 

Eq, (126). 


2 


38 4 

^, (125). 



HO. remarks. — It will be seen that the greatest strains 
in the 1st and 2d eases are as 2 to 1 ; and the same ratio holds 
in the 4th and 5th cases ; but in the 9th and 10th the ratio is 



FLEXURE. ] £>9 

as 3 to 2. The maximum strains in Cases VII. and VIII. do 
not occur at the points of maximum deflection. 

Although the moment in the first case is to that in the 2d as 
2 to 1, yet the deflections are as 8 to 3 ; and in the 4th and 5th 
cases the deflections are as 8 to 5. 

A comparison of Cases IV. and IX. shows the advantage of 
fixing the ends of the beam. The same remark applies to Cases 
V. and X. In the former cases the strain is only one-half as 
great when the beam is fixed at the ends as when it is supported, 
and in the latter two-thirds as great. 

Other interesting results may be seen by examining the table. 

The following are the results of some experiments made by 
James E. Francis : 

Experiment 1. A bar of " common English refined " iron, marked "J crown 
K, best," 12 feet 2£ inches long-, mean width 1.535 inch, mean depth 0.367 
inch, was laid on the 4 bearings, and loaded at the centre of each span, so 
as to make the deflections the same, the weight at the middle span being 
82.84 pounds, and at each of the end spans 52.00 pounds. The deflections 
with these weights were as follows : 

At the centre of the middle span 0.281 inches. 

At the centre of the end spans. . . .0.275 and 0.284 inches, mean, 0.280 " 

The deflections of the 3 spans being, as nearly as practicable, the same, the 
middle span is in the condition of a beam ' ' fixed at both ends and loaded in 
the middle," each of the end spans " being fixed at one end and supported at 
the other." A piece 3 feet ll-J inches long was then cut off from each end of 
the bar, leaving a bar 4 feet 4f inches long, which was replaced in its former 
position and loaded with the same weight (82.84 pounds) as before, when its 
deflection was found to be 1.059 inch, or 3.77 times the deflection when 
"fixed at both ends and loaded in the middle." 

Experiment 2. A bar of iron of the same quality and length as in Experi- 
ment 1, nearly square, its mean width being 0.553 inch, and mean depth 
0.549 inch, was laid on the same bearings, and loaded with the same weights, 
the deflections being as follows : 

At the centre of the middle span 0.342 inch. 

At the centre of the end spans 0.238 and 0.244 inch, mean, 0.241 " 

The bar was then reduced in length as in Experiment 1, leaving 4 feet 3| 
inches, which was replaced in its former position and loaded with the same 
weight (82.84 pounds) as before, when its deflection was found to be 0.983 
inch, or 4. 06 times the deflection, 4 1 when fixed at both ends and loaded in the 
middle." 

The result of both experiments agreed substantially with the deflection in 
9 



130 



THE RESISTANCE OF MATERIALS. 



the case of a beam "fixed at one end, supported at the other, and loaded in 



the middle," which is— 



0.438 of the deflection in the case, " supported at 



each end and loaded in the middle." In the foregoing experiments, the end 
spans correspond to this case, and the observed deflections with a weight of 52 
pounds, were 0.419 and 0.391 respectively, of the deflections in the case, 
"supported at the end and loaded in the middle," differing somewhat, but not 
very widely, from the proportion given above. 



111. PROBLEM. A PRISMATIC BEAM RESTS ON A SUPPORT 

AT THE MIDDLE OF ITS LENGTH, AND BARELY COMES IN CONTACT 
"WITH SUPPORTS WHICH ARE PLACED AT EACH END. SUPPOSE THAT 
AN UNIFORM LOAD IS PLACED ON ONE-HALF OF THE BEAM; IT IS 
REQUIRED TO FIND THE WEIGHT P WHICH, IF PLACED AT THE 
END WHICH IS REMOTE FROM THE UNIFORM LOAD, WILL CAUSB 
THE END TO WHICH IT IS APPLIED TO REMAIN IN CONTACT WITH 
THE SUPPORT. 



i/Wl-P i/ 2 Wl+21> 




A ^S 



Fig. 49. 



In Fig. 49, 

Letl = AB = BC; 

w = the load per foot of length on BO; and 
P = the weight at A which is necessary in order to keep 
the end down to the support. 

Take the origin at A, x horizontal and y vertical. 

Since no part of P is supported by A, it must be balanced 
by a part of the reaction of the support at C. 

The supports B and C each sustain one-half the uniform 
load; hence, 

%wl — P will be the reaction of the support O; 
\uol + 2P will be the reaction of the support B. 



FLEXURE. 131 

First consider the curve BO, and while so doing suppose that 
the part AB is rigid ; in other words, that the weight P does 
not cause AB to bend while the part BO is elastic. We then 
have for any point a> between B and C. 

Px — the moment of P ; 

(iwl + 2P) (x — l) = the moment of the reaction at B, 

which will have an opposite sign 

to Px; 
w(x—l) = the load on Ba ; 
\w{x— If = the moment of the load on Ba. 

Hence, 

(a) EI^ = Px + %w{x-lf-{^wl + 2P) {x-l) 

/. EI^ = %prj + iw (x-lf-l (wl -f 4P) (x-l)*+ 1 

also, Ely^Prf+^w { x -lf-^ M+4P) (x-l)* + O x x-Y 2 

But y = f or a? = I ; and 

^ = for a? =2£ ; which values in the last equation give 

= iPP+C 1 l+O 2 

= Ap^+^V^ 4 - -^(wl+^P) P+2CJ+ <7 2 = 
, PP- 1 frwl 4 +20 1 l+0 2 . 

Eliminating successively 6^ and 2 from these equations and 
we have, 

Ci = %PP^wlK 
These substituted in the preceding equations give 

(b) EI^jjL =iPx>+ {\wx - T %wl-P) (x-lf-iPP+^wF 

(c) Ely = iPx*+ (J T wx-lwl-iP) (x-lf-^PPx 

These equations will enable us to determine all the properties 
of the curve BO. But the solution of the problem only makes 
it necessary to find the inclination at B. For this x=l and we 
find 



132 THE KESISTANCE OF MATERIALS. 

Now consider AB as flexible, and we have 
dy _ Pv? 

-dx-2m + c * 

But this value of the tangent when x = lis the same as the 
preceding value 

.-. <7 8 =(-20P+ OT ?)^= 

Integrating again gives 

y = (4iV - 20i% + viM^^j + £4 

But by the conditions of the problem 

y = for x = .\ <7 4 = 

Also, 2/ = for a? = I , 

... o = - 16PZ 3 + wZ 4 , 

This problem was suggested by the conditions of the draw- 
bridge. If the end is not held down, the distance which it will 
raise by an uniform load on the other half is found from Eq. 
(d), by making P == 0, and multiplying by I. This will give 

24cFI 



113. TO FIND THE REACTION OF THE SUPPORTS, 

we may first find the bending moments over the supports, according to Clapey* 
"on's method. 



Fig. 49a. 



FLEXTJEE. 133 

Let A, B and G be any three consecutive supports ; 
l f the segment AB, and I' the segment BG ; 
w the pressure on a unit of length on AB, and w' on BG\ 
X lt X, 2 X 3 the bending moments at A, B, and <7, respectively; 

Take the origin at A. The moment of external forces upon any point in the 
segment AB, will involve the moment of all the reactions at the left of A, 
and the moment of the total load to the left of A, plus the moment of the 
load on x. The moments of the two former may contain the first power of x, 
and possibly they may also contain a constant. Let A and B be constants; 
then the equation for the moment of flexure will be 

(a) EI^ = A + Bx-^wx 2 

If x = 0, the second member becomes 

A = X U 

and if * = I, we have A + Bl — %wl 2 — X 2 

(5) . \ EI^L = X l + X *~ Xl x+\wlx-\wx* 
similarly, if the origin be taken at B we have 

(e) El P{ = x 2 +^_Z— 2 x + Wte - W&. 



Integrating Eq. (5), observing that — == tang i' for x = 0, and tang *'" for x = 

I and y = for x = and for # = I Between the equations thus formed eli- 
minate tang i\ and find the value of tang i" from Eq. (c), and substitute 
its value in the preceding. This done and the result may be reduced to the 
following form : — 

(d) X x l+2X 2 {I t-l') + X3l'+l,(wP+w'V*)= 

which expresses the relation between the bending moments at any three con- 
secutive points of support. 

By applying this equation successively to the successive points, the bending 
moments at all the points of support may be found, after which the bending 
moments of any point of any of the segments by Eq. (a) or (b). The reactions 
may also be easily found by the aid of the results. 

If I = I' = &c, and w =w" = &c, Eq. (d) becomes 

(e) X 1 +4X 2 +X 3 +&rt*=Q J 
and for the second, third, and fourth supports we have 

X 2 +4X, 4- X 4 + iwl* = 0, 

and so on. By taking the difference between these, we find the relation be- 
tween the bending moments for four consecutive points of support indepen* 
dently of the uniform load. 






134 THE RESISTANCE OF MATERIALS.. 

Example. — Suppose that there are five points of support, equidistant; and 
the load uniform. 

The bending moment at the first support, A, Fig. 49a, is zero ; that at B 
equals that at D, — supposing that there is a fifth point, £!, beyond D. Henoe 
for the first two segments, Eq. (e\ gives 

0+4X 2 +X, + iwZ 2 = 

and for the second and third segments ; 

X 2 +4:X 3 + X 2 +^wP = 0, 

orX 2 +2X 3 +i^ 2 = 0; 

and by elimination 

X 2 = —g-gwl 2 , and X 3 = — fowl 2 . 

Let P L , P 2 , &c, be the reactions of the supports at J., B, &o., then the 
moments at B are 

P x Z- \wl 2 = X 2 =z- -£$wl 2 

where W= the load on each segment. 
For the moments at G we have 

P 1 .2l+P 2 l-wl^l-wl : kl=^i=—h^l 2 
..\Pi=W>l- 
The total load is 

2P 1 +2P 2 +P 6 =4wl 
.-.P 3 =fW, 

which are the same as those given in the table on page 135. 
Whatever be the number of props, we have for the first segment 

El^=-V l x+\wx 2 

eA=-\Y x x*+\wx*+C x 
ax 



'= -i V x x 3 + J-fiox* -t- dx + G 2 
For x=0, y=0 ; also for x—l, y=0 

.-. C 2 =0, and d^VJ 2 — *hwl 3 . 
Hence over the first support we have 

and for the deflection at the middle of the first segment 

J=(24r, -7 W)^; 

which is always somewhat less than the maximum deflection, except when the 
beam is supported at its ends only. • 



FLEXURE. 



135 



ADDITIONAL PROBLEMS WHICH ARE PURPOSELY LEFT 

UNSOLVED. 

1. Suppose that a beam is supported at its extremities, and has two forces 
at any point between. In this case the curve between the support and the 
nearest force will have one equation ; the curve between the forces another ; 
and the remaining part a third. 

2. In the preceding case, if the forces are equal and equidistant from the 
supports, the curve between the forces will be the arc of a circle. 

3. Suppose that the beam is uniformly loaded and rests on four supports. 

4. Suppose that the beam is supported at its extremities and has a load uni- 
formly increasing from one support to the other. 

5. Suppose that the beam is uniformly loaded over any portion of its 
length. 

6. Suppose that it has forces applied at various points. 
These problems will suggest many others. 

7. Suppose that a beam is supported at several points, and loaded uniformly 
over its whole length. 

Let W= the weight between each pair of supports, • 

V\, Vi, V-it & c -> he the reactions of the supports, counting from one 
end, 
and let the distances between the supports be equal. 
Then we have : — 



No. of 

Sup- 
ports. 


V x 


v 2 


2 


iW 


w 


3 


t 


Ifl 
8 


4 


A 


lo- 


5 


tt 


ll 


6 


L5. 

38 


A3. 

38 


7 


-Al- 
io 4 


HI 


8 


* 


m 


9 


m 


4.4.0 
3 88 


10 


2JL9, 
53$ 


BJLL 

530 



r. 



Frac 
I 
\h 
ft 

II 

m 



#3 



V, 


v s 


r 6 


v 7 


F 8 


v 9 


tional 


parts 




oiW. 






A 












II 


tt 










h 


ft 


if 








m 


m 


HI 


■N* 






m 


m 


m 


m 


* 




m 


tit 


3.3.2. 

38 8 


5.X4 

388 


m 


m 


SA&. 

630r 


HI 


£.aa 

630 


5.3.5. 
530 


&h 


m 



F 10 



m 



If the beams and props were perfectly rigid, all but the end ones would sus- 
tain TF, and the end ones each \ W. 



136 THE RESISTANCE OF MATERIALS. 

113. BEAMS OF VARIABLE SECTIONS. 

For these /is variable, and its value must be substituted in 
Equation (50) before the integration can be performed. As an 

example, let the beam be 
— L -rS) fixed at one extremity, and 

^___ I ■ \ a weight JP, be suspended at 

I the free extremity. Fig. 50. 

._ ; __ m ___ f Let the breadth be constant, 

- , ' , L ) and the longitudinal vertical 
^sL I i ' \i section be a parabola. Then 

pjjjl all the transverse sections 

^^ F 50 will be rectangles. 

Let I = the length, 

b = the breadth, and 
d = the depth at the fixed extremity. 
If y i is the whole variable depth at any point, we have, from 
the equation of the parabola, 

d 2 
(faff = px, or \d* =j?l, .'.jp = j7, in which p is the parameter 

of the parabola. 

.-.jf-T—- (m) 

From Equation (51) we have 

I— -fehy?, in which substitute^, from Equation (127), and wo 

have /= T V^* (128) 

The equation of moments is, see Equation (50), 

EI -t^j = jPx, in which substitute I, from Equation (128), 
and we have 

dx>~ Ebd? x 2 
Multiply by dx and integrate, observing that -jj- = f or x — 1 



FLEXURE. 



137 



and we have 



dy_ 
dx 



24:Pi$ 



EM* ^-*> 

Integrating again gives 



y 



EM^ x ~ Slx) 

y is zero for x = 0. 
y = A for x = I ; 

8PZ 3 



\A 



Eld 3 



.(129) 



If, in Equation (57), we substitute 7= ^M s (Eq. (51) ), it be- 
comes 

A ~~ Ebd? 

which is one-half that of Eq. (129) ; hence the deflection of a pris- 
matic beam is one-half that of a parabolic beam of the same 
length, breadth, and greatest depth, when fixed at one end and 
free at the other, and has the same weight suspended at the 
free end. 

In a similar maimer the equation of the curve may be found 
for any other form of beam, if the law of increase or decrease 
of section is known. Several examples may be made of beams 
of uniform strength, which will be given in Chapter VII. 

114. BEAMS SUBJECTED TO OBL.IQUE STRAINS. 

Let the beam be prismatic, fixed atone end, 
and support a weight, P, at the free end ; 
the beam being so inclined that the direction 
of the force shall make an obtuse angle with 
the axis of the beam, as in Fig. 51. 

Let Pj = P sin 6 = component of P per- 
pendicular to the axis of the beam, 
and 
P 2 — P cos 6 = component parallel to 
the axis of the beam. 
Take the origin at the free end, the axis 
of x being parallel to the axis of the beam, 
and y perpendicular to it. 

Then Equation (50) becomes FlG 




138 



THE RESISTANCE OF MATERIALS. 



El^ 2 =^P x x+P 2 y 



°*,^ = -P 2 *+Q 2 &- 



(130) 



p 

in which p 2 =^>5 ^ q 2 = ~- r The complete integral of (130) is (see Ap 



Pi 

ev 



pendix). 



El' 



gas -gx p 2 

y = C 1 e + C 2 e +p& 



The conditions of the problem give 

dii 

= for x = l\ and 



dx 
y = for x = ; and these combined with the preceding equatioE 



give:— 



(ql —ql\ «2 

C x e - C 2 e j + ^- 2 ; 



= ^ + C 2 ; 

From which C x and G 2 may be found, and the equation becomes completely 
known. 
We also have y — J for x = I; 

ql —ql p2 

.*. J= O + C 2 e +- 2 l', 

Next, suppose that the force makes an acute angle with the axis of the beam, 
as in Fig. 52. 

For the sake of variety, take the origin at A, the fixed end, x, still coin- 
ciding with the axis of the beam before flexure. Using the same notation as 
in the preceding and other cases, we have 



^l 2 =p\i-x)+q 2 u-y) 



(131) 



The complete integral is 

J-y = Asmq{x + B)-^ Y (l^x) (132) 

in which A and B are arbitrary constants. 
From the problem we have 
y = f or x '— ; 

4- — f or x = ; and 
dx ' 

y — A f or x = I ; 
by means of which the equation becomes 
Fig. 52. completely known. 

One difficulty in applying these cases in practice is in determining the value 




FLEXUEE. 



139 



of I. Before it can be determined, the position of the axis mnst be known. 
According to Article 78, 3d case, it appears that the neutral axis does not co- 
incide with the axis of the beam. Indeed, according to the same article, it ia 
not parallel to the axis, and hence / is variable, and the equations above are 
only a secondary approximation ; the first approximation being made in estab- 
lishing Equation (50), and the next one in assuming /constant. In practice 
we assume that / is constant for prismatic beams, and that the neutral axis 
coincides with the axis of the beam. 



115. FLEXURE OF COI/UOTNS. 

If a weight rests upon the axis 
of a perfectly symmetrical and 
homogeneous column, we see no 
. reason why it should bend ; but 
in practice we know that it will 
bend, however symmetrical and 
homogeneous it may be, and how- 
ever carefully the weight may 
be placed upon it. If the weight 
be small, the deflection may not be 
visible to the unaided eye. If the 
weight is not so heavy as to crush 
the column, an equilibrium will 
be established between the weight 

and the elastic resistance within the beam. Let the column rest 
upon a horizontal plane, and the weight P on the upper end be 
vertically over the lower end. Take the origin of coordinates 
at the lower end of the column, Fig. 53, a; being vertical, and y 
horizontal. They must be so taken here, because x was as- 
sumed to coincide with the axis of the beam when Equation 
(50) was established. Then y being the ordinate to any point 
of the axis of the column after flexure, the moment of P is Py 9 
which is negative in reference to the moment of resisting forces, 
because the curve is concave to the axis of a?, in which case the 
ordinate and second differential coefficient must have contrary 
signs (Dif. Cal.). Hence we have, 




Fig. 54. 



*3~-* 



(133) 



140 THE RESISTANCE OF MATERIALS. 

Multiply by dy and integrate (observing that dx is constant) v 
and find 

But ^— = f or y = A = the maximum deflection. These 

PA 2 
values in the preceding equation give C x = -j^f, which beinp 

substituted in the same equation and reduced gives 



dx 



=V! 



EI dy 



P VA 



I ei • - 1 y 



y 

sin -4- + C % . 

But y = for a? = .*. (7 2 = 0. Hence the preceding gives 

y^AAu^J^x (134) 

But y == for x = 1. Therefore, if n is an integer, these 
values reduce (134) to 



\Jm* latim * 



n 2 7z 2 



.:P = EI- ir (135) 

This value of P reduces (134) to 

y = A sin nn j 

which is the equation of the curve. It is dependent only upon 
the length of the column and the maximum deflection. If 
n = 1, the curve is represented by a, Fig. 54 ; if n = 2, by b ; 
if n = 3, by o. 

If n = 1, Equation (135) becomes 

P=^E1 (136) 



FLEXURE. 141 

which is the formula to be used in practice. "We see that the 
resistance is independent of the deflection. If the column is 
cylindrical, I = J n r 4 (see Equation (52) ) ; 

.^«V x ? (137) 

hence the resistance varies as the fourth power of the radius 
(or diameter), and inversely as the square of the length. If the 
column is square, 1=^^ (Equation (51)), 

•'' F =T2 X J ( 138 > 

These formulas, according to Navier* and "Weisbach,f should 
be used only when the length is 20 times the diameter for 
cylindrical columns, or 20 times the least thickness for rectan- 
gular columns ; and INavier says that for safety only ^ of the 
calculated weight should be used in case of wood, and J to ^ in 
case of iron ; but Weisbach says they should have a twenty-fold 
security. 

Examples. — 1. What must be the diameter of a cast-iron column, whose 
length is 12 feet, to sustain a weight of 30 tons (of 2,000 lbs. each) ; E = 16,- 
000,000 lbs. ; and factor of safety ^. Ans. d = 7.52 in. 

2. If the column be square and the data the same as in the preceding ex- 
ample, Equation (138) gives 



4 / 12 x 60,000 x (12 x 12)* x 20 a _ . , 
V (3.1416)* x 16,000,000 = b ' 6 mche *- 



In the analysis of this problem I have followed the method 
of Navier ; but practical men generally prefer the empyrical 
formulas of Article 62. But it will be observed that the law 
of strength, as given in the formulas in that article, are the 
same as those given in equations (137) and (138) for wooden 
columns, and nearly the same as for iron ones. The chief dif- 
ference is in the coefficients, or constant factors. In the analy- 
sis it was assumed that the neutral axis coincides with the axis 
of the beam, but it is possible for the whole column to be com- 
pressed, although much more on the concave than on the con- 

* Navier, Resume des Lecons, 1838, p, 204 

t Weisbach's Mechanics and Engineering. Vol. 1, p. 219. 1st Am. ed. 



142 THE RESISTANCE OF MATERIALS. 

vex side, in which case the neutral axis would be ideal, having 
its position entirely outside the beam on the convex side. In 
this case, if the ideal axis is parallel to the axis of the beam, 
the value of /will be constant; and equations (137) and (138) 
retain the same form. The problem of the flexure of columns 
is then more interesting as an analytical one than profitable as 
a practical one. 

GRAPHICAL METHOD. 

116, the graphical method consists in representing 
quantities by geometrical magnitudes, and reasoning upon them, 
with or without the aid of algebraic symbols. This method has 
some advantage over purely analytical processes ; for by it 
many problems which involve the spirit of the Differential and 
Integral Calculus may be solved without a knowledge of the 
processes used in those branches of mathematics ; and in some 
of the more elementary problems, in which the spirit of the 
Calculus is not involved, the quantities may be directly pre- 
sented to the eye, and hence the solutions may be more easily 
retained. It is distinguished, in this connection, from pure 
geometry by being applied to problems which involve mechan- 
ical principles, and to use it profitably in such cases requires a 
knowledge of the elementary principles of mechanics as well 
as of geometry. 

But graphical methods are generally special, and often re- 
quire peculiar treatment and much skill in their management. 
It is not so powerful a mode of analysis as the analytical one, 
and those who have sufficient knowledge of mathematics to use 
the latter will rarely resort to the former, unless it be to illus- 
trate a principle or demonstrate a problem for those who cannot 
use the higher mathematics. A few examples will be given 
to illustrate this method. 

117. GENERAL PROBLEM OF THE DEFLECTION OF 

beams. — To find the total deflection of a prismatic beam 
which is bent by a force acting normal to the axis of the beam 
without the aid of the Calculus. 

Let a beam AB, Fig. 55, be bent by a force, P, in which 



FLEXUEE. 



143 



case the fibres on the convex side will be elongated, and those 
on the concave side will be compressed. Let AB be the neu- 
tral axis. Take two sections normal to the neutral axis at L 




Fig. 55. 



and N, which are indefinitely near each other. These, if pro- 
longed, will meet at some point as 0. Draw KN parallel to 
LO. Then will he, = X, be the distance between iT-ZT and EN 
at k, and is the elongation of the fibre at h. Let eN '= y, then 
from the similar triangles JcNe and ZONwe have 



ON: New LIST: he 



Ne. LN LN 



ON 



ON 



If, now, we conceive that a force ^>, acting in the direction of 
the fibres, or, which is the same thing, acting parallel to the 
axis of the beam, is applied at k to elongate a single fibre, we 
have, from Equation (3) and the preceding one, 



P 



— — Ice 

EAa LN- 



E T 



in which A a is the transverse section of the fibre. As the sec- 
tion NK turns about N on the neutral axis, the moment of this 
force is 

jp 

which is found by multiplying the force by the perpendicular y. 



144 THE RESISTANCE OF MATERIALS. 

This is the moment of a force which is sufficient to elongate 
or compress any fibre whose original length was ZN, an amount 
equal to the distance between the planes iTiV" and JSTVmeasured 
on the fibre or fibre prolonged. Hence, the sum of all the mo- 
ments of the resisting forces is 

Spy = -fift Zy* J<* 

in which 5 denotes summation ; and in the first member means 
that the sum of the moments of all the forces which elongate 
and compress the fibres is to be taken ; and in the second mem- 
ber it means that the sum of all the quantities tf Aa included 
in the transverse section is to be taken. The quantity, Sy* Aa 
is called the moment of inertia, which call I. 

But the sum of the moments of the resisting forces equals 
the sum of the moments of the applied forces. Calling the 
latter SPX, in which X is the arm of the force P, and we have 

Spy = 2PX= ojy-2y > Aa = ~^- 

■'■ 0N= YFX (139) 

In the figure draw Zb tangent to the neutral axis at Z, and 
Na tangent at N. The distance ab, intercepted by those tan- 
gents on the vertical through A, is the deflection at A due to 
the curvature between L and N. As ZJV is indefinitely short, 
it may be considered a straight line, and equal x ; and Zb—ZO 
very nearly for small deflections ; and ZG = X. (Z stands for 
two points.) 

By the triangles OZJy and aZb, considered similar, we have 

ON : x : : Zb : ab = -yv> 

in which substitute ON from Equation (139) and we have 

. XxXPX 
ah =—£J— ( 14 °) 

which is sufficiently exact for small deflections. If, now, tan- 
gents be drawn at every point of the curve AB, they will divide 



FLEXURE. 145 

the line A C into an infinite number of small parts, the sum of 
which will equal the line A 0, the total deflection. But the 
expression for the value of each of these small spaces will be of 
the same form as that given above for ah, in which P, E, and I 
are constant. 

This is as far as we can proceed with the general solution, 
"We will now consider 

PARTICULAR CASES. 

118. CASE 1. L.ET THE BEAM BE FIXED AT ONE END, 
AND A LOAD, P, BE APPLIED AT THE FREE END._This 
is a part of Case I., page 109, and Fig. 37 is applicable. The moment of P, 
in reference to any point on the axis, is PX. Hence 2PX is simply PX, 
which, substituted in Equation (140), gives 

ab = ^X 2 x 

.'.AC=-?- 1 2X*x (141) 

This equation has been deduced directly from the figure. It now remains 
to find the sum of all the values of X' 2 x, which result from giving to Xall pos- 
sible values from X = to X = I * To do this, construct a figure some prop- 
erty of which represents the expression, but which has not necessarily any 
other relation to the problem which is being solved. If X be used as a linear 
quantity, X' 2 may be an area and X' 2 x will be a small volume. These condi- 
tions are represented by a pyramid, Fig. 56, in which 

AB = I = the altitude, and the base BCDE is a square, whose sides, BC 
and CD, each = I. Let bode be a section parallel 
to the base, and make another section infinitely 
near it, and call the distance between the two sec- 
tions x. 

Then Ab = X --= be — cd, 

X 2 = area bcde, and 
X 2 x = the volume of the lamina 
bcde, 
which is the expression sought. The sum of all 
the laminas of the pyramid which are parallel to 
the base is limited by the volume of the pyramid, 
and this equals the value of the expression 2X 2 « f IG# $q 

between the limits and I. The volume of the 

pyramid is the area of the base (— I 2 ) multiplied by one-third the altitude 
(s0> or \P, which is the value sought. 

* This by the Calculus becomes fx 2 dx = %F. 
10 




146 THE RESISTANCE OF MATERIALS. 

Hence, AC - ^-j. 

which is the same as Equation (57). 

The value of X 2 x may also be found by statical moments as follows : — Let 
ABC, Fig. 57, be a triangle, whose thickness is unity, and 
which is acted upon by gravity (or any other system of paral- 
lel forces which is the same on each unit of the body). Take 
an infinitely thin strip, be, perpendicular to the base, and 




Ab = X = be, and 
Fig. 57. V — the weight of a unit of volume. 

Then Xx = the area of the infinitely thin strip be, and 
pXx = the weight of the strip be, and 
pX' 2 x = the moment of the strip, when A is taken as 
the origin of moments. If the weight of a unit of volume be taken as a unit, 
the moment becomes X 2 x, which is the quantity sought, and the value of 
2X 2 x from to Ms the moment of the whole triangle ABC. Its area is -^Z 2 , 
and its centre of gravity \l to the right of A. Hence the moment is %l s as 
before found.* 

119. CASE II.— LET THE BEAM BE FIXED AT ONE END, 

AND UNIFORMLY LOADED OVER ITS WHOLE LENGTH.— 

This is the same as a part of Case II., page 110, and Fig. 39 is applicable. 

Let Xbe measured from the free end, and 
w = the load on a unit of length ; then 
wX = the load on a length X, and 

\X — the distance of the centre of gravity of the load from the section 
which is considered. 
Hence the moment is \wX' 2 , which equals 2PX, and Equation (140) becomes 

ab = 2ih XiXi and 

w k=i 
AC — oeTT 2X 3 ic = the total deflection. 

To find the value of 2X ? #, observe, in Fig. 56, that X 2 x is the volume of 
the lamina bede, and this multiplied by the altitude of A — bede, which is X, 
gives X 3 x, the expression sought. Hence the sum sought is the volume of the 
pyramid A — BCDE, multiplied by the distance of the centre of gravity of 
the pyramid from the apex ; or, 

. wl* _ Wl* 

' - Aa -8RI-8Kr (U3) 

where W is the total load on the beam. 



* This may be written 2 X?x = \V\ 
«=0 



FLEXTTKE. 147 

120. CASE III.-LET THE BEAM BE SUPPORTED AT 
ITS ENDS AND LOADED AT THE MIDDLE BY A WEIGHT P, 

as in Fig. 40. The reaction of each support is ^P, and the moment is ^PX, 
and Equation (140) becomes 

But in this case the greatest deflection is at the middle, and the limits of 
2X 2 x are and \l. Hence, in Fig. 56, let the altitude of the pyramid be \l, 
and each side of the base also \l, and the volume will be 

ilxilxi of 11 = ^1* 

PP 



.'.AG = 



4S&.r 

which is the same as Equation (73). 



131. CASE IV. LET THE BEAM BE SUPPORTED AT ITS 
ENDS AND UNIFORMLY LOADED, AS IN FIG. 42. 

w being the load on a unit of length, the reaction of each support is \wl, 
and its moment at any point of the beam is \wlX. On the length X there is 
a load wX, the centre of which is at ^X from the point considered ; hence its 
moment is %wX' z , and the total moment is the difference of these moments; 

.♦. 2PX=iwlX-±wX 2 , 

and Equation (140) becomes 

ab = ^- I {lX*x-X*x), 

and the total deflection at the middle is, 



AG = -?-llLX*x- £X 3 A 
2E.l\ x=z0 x=0 ) 



The values of the terms within the parentheses have already been found, 
and by substracting them we have 

AG- 5 W 



139. REMARKS ABOUT OTHER CASES.— This method, which 
appears so simple in these cases, unfortunately becomes very complex in many 
other cases, and in some it is quite powerless. To solve the 9th and 10th 
cases, pages 124 and 125, necessitates an expression for the inclination of the 
curve, so that the condition of its being horizontal over the support may be 
imposed upon the analysis. But the 9th case may be easily solved if we find 
by any process that the weight which must be suspended at the outer end of 



148 



THE RESISTANCE OF MATERIALS. 



the beam to make it horizontal over the support is \Pl divided by AG, Fig. 
47. For the reaction of the support is \P+P X ; 

.-. PX=P 1 (AG+X)-$P+P 1 )X 

=P X AG-\PX 

=\Pl-\PX 

PIXx - ±PX 2 x 



ab = i 



B.I 



and the deflection at the centre = i -=r- 7 {IZXx — 42X 2 -#) taken between the 

Mi. 1 

limits and \l. 

The part YlXx is the area of a triangle whose base and altitude are each £?, 

.• . ZXx = & 2 , and 2X 2 a? between the limits and \l, is &1 2 . \ AG (Fig. 55), 

P_ J*_ 
~ 192 B.I' 

All these expressions contain 7", the value of which remains to be found by 
the graphical method. 



123. MOMENT OF INERTIA OF A RECTANGLE. 

Required the moment of inertia of a rectangle about one end as an axis. 

Let ABGD, Fig. 58, be a rectangle. Make BG perpendicular to and equal 
AB, and complete the wedge G — ABGD. 
Let Aa = the area of a very small surface at E, 
and y — AE = BF, then 
yAa — the volume of a very small prism BF, 

and this multiplied by y gives 
^ 2 Aa=:the moment of inertia of the elemen- 
tary area at E, which is also the 
statical moment of the prism BF, 
and 

ly* Aa=I=the moment of inertia of the rec- 
tangle ABGD. 
Hence the moment of inertia of the rectangle is 
represented by the statical moment of the wedge 
G - ABGD. If 

AB=d = BG, and 
AD = b. 




then the volume of the wedge is 

bd x \d 
uid the moment = \bd z x \d = \bd z . . 



ibd* 



)143) 



If the axis of moments passes through the centre of the rectangle, and 



FLEXURE. 



14:9 



parallel to one end, we have BE — GB — \d in Fig. 59. Hence the moment 
of inertia of the rectangle = 

2xbx\dx\dx%oi \d = -hbd* 

which is the same as Equation (51). 

124. THE MOMENT OF INERTIA OF A TRIANGLE about 
an axis parallel to the base and passing through the vertex is, in a similar 
way, the statical moment of the pyramid ABODE, Fig. 60. 

Let b — CB= base of the triangle, and 

d — AB — BD = GE = altitude of the triangle and pyramid and sides 
of the base of the pyramid. 

The volume of the pyramid = \bd 2 . 

The centre of gravity is $d from the apex, consequently the statical moment 
i&\bd z xid = \bd A . 

But in a triangular beam the neutral axis passes through the centre of grav- 
ity of the triangle, and it is desirable to find the moment of inertia about an 
axis which passes through the centre and parallel to the base. 

This may be done as in the preceding Article ; but it may be more easily 





done by using the formula of reduction, which is as follows : — The moment of 
inertia of a figure about an axis passing through its centre equals the moment 
of inertia about an axis parallel to it, minus the area of the figure multiplied by 
the square of the distance between the axes. (See Appendix. ) 

This gives for the moment of inertia of a triangle about an axis passing 
through its centre and parallel to the base 



\bd* - \bd x (P) 2 = -h bd* 



.(143) 



125. THE MOMENT OF INERTIA OF A CIRCLE may be 

represented in the same way, but it is not easy to find the volume of the 
wedge, or the position of its centre of gravity, except by an analysis which is 
more tedious than that required to find the moment directly, as was done in 
Equation (51). But it may be found practically, by those who can only per- 



150 



THE RESISTANCE OF MATERIALS. 



form multiplication, as follows : — Make a wedge-shaped piece out of wood, 
or plaster-of-Paris, or other convenient material, the base of which is the sem- 
icircle required, and whose altitude equals the radius of the eircle, as shown 
in Fig. 60a ; then find its volume by immersing it in a liquid and measuring 
the amount of water displaced. Then determine the horizontal distance to 
the centre of gravity of the wedge from the centre of the circle by balancing 
it on a knife edge, holding the edge of the knife under the base of the wedge, 
and parallel to the edge, ab, of the wedge, keeping the side vertical, and meas- 
uring the distance between the edge ab and the line of support. Then the 
statical moment of the wedge, which equals the moment of inertia of the 
semicircle, is the product of the volume multiplied by the horizontal distance 
of the centre from the edge, and twice this amount is the moment of inertia 
of the whole circle. Its value for the whole circle, or for both wedges, is 

There are, however, many methods of calculating the moment of inertia of 
a circle without using the Calculus. The following appears as simple as any 
of the known methods : — 

The moment of inertia of a circle is the same about all its diameters. Hence 
the moment about X in the figure, plus the moment about Y, equals twice the 
moment about X. The distance to any point A is p, and equals V X 2 + y 2 ; 
or p 2 = x 2 + y 2 ; and if Aa be an elementary area, as before, we have 

22Aa x 2 = 2Aa x 2 + 2Aay 2 = 2Aap 2 , 




Fig. 60a 





Fig. 60c. 



Fig. 605. 



the latter of which is called the polar moment of inertia, in reference to an 
axis perpendicular to the plane of the circle, and passing through its centre C. 
To find the value of 2Aa p 2 , take a triangle whose base and altitude are each 
equal to r, the radius of the circle, and revolve it about the axis through C, 
and construct an infinitely small prism on the element Aa as a base. 

We have p = CA = AB, Fig. 60c. 

Aa p = volume of the small prism AB. 

Aap GA = Aa p 2 = the statical moment of AB, 

which expression is of the form of the quantity sought. 

Hence 2Aa p 2 is the product of the volume of the solid generated by the trian- 
gle, multiplied by the abscissa of its centre of gravity from C. The solid is 
what remains of a cylinder after a cone has been taken out of it, the base of 



FLEXURE. 151 

the cone being the upper base of the cylinder, and the apex of which is at the 
centre of the base of the cylinder. Hence the volume of the solid is the vol- 
ume of the cylinder, less the volume of the cone ; 

or 7rr 2 x r— -rrr 2 x \r =§7rr 3 . 

If now the solid be divided into an infinite number of pieces, by planes 
which pass through its axis, each small solid will be a pyramid, having its 
vertex at (7, and the abscissa to the centre of gravity of each will be \r from 
G. Hence we finally have 

ZAa p 2 = f ttt* 3 x fr = ^r 4 , 
which equals 22A# x 2 . 

.-. 2Aa« 2 =i7rr 4 (144) 



126. MOMENT OF INERTIA OF OTHER SURFACES. — 

The general method indicated in the preceding articles is applicable to sur- 
faces of any character, and with careful manipulation approximations may be 
made which will be very nearly correct, and, as we have seen above, in some 
cases exact formulas may be found. 



127. vibrations OF BEAMS— If a load be placed suddenly 
upon a beam, and be left to the action of the elastic forces, it will vibrate. 
Or if a load is upon the beam and the deflection be increased or decreased by 
an external force, and then left to the action of the elastic forces, it will 
vibrate the same as before. Take the case shown in Fig. 36, and suppose 
that the weight is applied suddenly. 

Let z be the variable deflection ; then from Eq. (57) we find that the pres- 
sure, P', which will produce this deflection is 

and hence the pressure which is still available for producing the maximum 
deflection is 

d 2 z P d 2 z 

From Mechanics we have ■=-= = the acceleration and — ~rrs= the moving pres* 
dt 2 g dv 

sure = -^- {4-z). 



Integrating once gives 



P dz 2 SEI /0 . gx 



dgBI V2Az-z* 



152 THE RESISTANCE OF MATERIALS. 



For g = A. we have 



•••«-'/ 



« . - f 



*=*Jmr*Si> 



Sgffl 

hence they are isochronous. The weight of the beam has been neglected. 
We would find a similar expression if the beam were uniformly loaded, or if 
supported at its ends. 



TEANSVEESE STEENGTH. 



153 



CHAPTER VL 



TRANSVERSE STRENGTH. 



128. STRENGTH OF RECTANGULAR REAMS. The tne- 

ories which have been advanced from time to time to explain 
the mechanical action of the fibres, have been already given in 
Chapter IV. 

First, consider the common theory, according to which the 
neutral axis passes through the centre of gravity of the trans- 
verse sections, and the strain upon the fibres is directly propor- 
tional to their distance from the neutral axis. 

Continuing the use of the geometrical method, let Fig. 61 
represent a rectangular beam 
which is strained by a force P 
applied at any point. Let de be 
on the neutral axis, and ah repre- 
sent the strain upon the lowest 
fibre. Pass a plane, de-cb, and 
the wedge so cut oif represents 
the strains on the lower side, and 
the similar wedge on the other 
side represents- the strains on the 
upper side. 

Let It = the ultimate strain 
upon a unit of fibres most re- 
mote from the neutral axis on the side which first ruptures, on 
the hypothesis that all the fibres of the unit are equally 
strained, and b = the breadth and d — the depth of the beam. 

Let ab — R\ then, the total resistance to compression — \Rb 
x \d — \Rbd, = the volume of the lower wedge ; and the mo- 
ment of resistance is this value multiplied by the ordinate to 
the centre of gravity of the wedge froin de, which is f of \d = 
\d\ consequently the moment is 




Fig. 61. 



154 THE RESISTANCE OF MATERIALS. 

■&Bb<P; 

and as the moment of resistance to tension is the same, the total 
moment of resistance is 

iBbcP (145) 

which equals the moment of the applied or bending forces. 

If the beam be fixed at one end and loaded by a weight, P, 
at the free end, we have for the dangerous section, or that most 
liable to break, 

PI = \BW. 

In rectangular beams the dangerous section will be where the 
sum of the moments of stresses is greatest, the maximum values 
of which for a few cases are given in a table on page 128. 
Using those values, and placing them equal to \Bbd n , and we 
have for solid rectangular beams at the dangerous section, the 
following formulas : — 

for a beam fixed at one end and a load, p, at the fkee 
end; 

PI = \Bbd? (146) 

And foe an uniform load ; 

\Wl = \Bld? (147) 

Foe a beam supported at its ends and a load, P, at the 
middle; 

\Pl = \BW (148) 

And foe an uniform load ; 

\m=\Bl& : . . ..(149) 

And for a load at the middle, and also an uniform load : 

i(2P + W)l =\Bbd 2 (150) 

For a beam fixed at both ends and a load, P, at the 
middle; 

iPl = \Bbd? (151) 

And for an uniform load, end section ; 

T ^Wl = \Bb& (152) 

Middle section ; 

frWl = \BW (153) 



TRANSVERSE STRENGTH. 155 

These expressions show that in solid rectangular beams, the 
strength varies as the breadth and square of the depth, and 
hence breadth should be sacrificed for depth. In all the cases, 
except for a beam fixed at the ends, it appears that a beam will 
support twice as much if the load be uniformly distributed over 
the whole length as if it be concentrated at the middle of the 
length. The case in which a beam is fixed at both ends and 
loaded at the middle lias given rise to considerable discussion, 
for it is found by experiment that a beam whose ends are fixed 
in walls of masonry will not sustain as much as is indicated by 
the formula, and also that it requires considerably more load to 
break it at the ends than at the middle, but the analysis shows 
that it is equally liable to break at the ends or at the middle. 
But it should be observed that there is considerable difference 
between the condition of mathematical fixedness, in which case 
the beam is horizontal over the supports, and that of embedding 
a beam in a wall. For in the, latter case the deflection will 
extend some distance into the wall. 

Mr. Barlow concludes from his experiments that Equation 
(151) should be 

iPl = \Rbd? (154) 

and this relation is doubtless more nearly realized in practice 
than the ideal one given above. In either case, it appears that 
writers and experimenters have entirely overlooked the effect 
due to the change of position of the neutral axis, which must 
take place. It has been assumed that the neutral axis coincides 
with the axis of the beam, and that its length remains unchanged 
during flexure ; but if the ends of the beam are fixed, the axis 
must be elongated by flexure, or else approach much nearer the 
concave than the convex side, or both take place at the same 
time, in which case the moment of resistance will not be \RbcP. 
The phenomena are of too complex a character to admit of a 
thorough and exact analysis, and it is probably safer to accept 
the results of Mr. Barlow in practice than depend upon theo- 
retical results. 



156 THE RESISTANCE OF MATERIALS. 



129. modulus of rupture. — When a beam is sup 
ported at its ends, and loaded uniformly over its whole length 
and also loaded at the middle, we find from Equation (150) 

*-«&=' m 

in which W may be the weight of the beam. Beams of known 
dimensions, thus supported, have been broken by weights placed 
at the middle of the length, and the corresponding value of It 
has been found for various materials, the results of which have 
been entered in the table in Appendix III. This is called the 
Modulus of Rupture, and is defined to be the strain at the 
instant of rupture upon a square inch of fibres most remote 
from the neutral axis on the side which first ruptures. It 
would seem from this definition that R should equal either the 
tenacity or crushing resistance of the material, depending upon 
whether it broke by crushing or tearing, but an examination of 
the table shows the paradoxical result that it never equals 
either, but is always greater than the smaller and less than the 
greater. 

The tabulated values of R being found from experiments 
upon solid rectangular beams, they are especially applicable 
to all beams of that form, and they answer for all others that do 
not depart largely from that form ; but if they depart largely 
from that form, as in the case of the X (double T) section, or 
hollow beams, or other irregular forms, the formulas will give 
results somewhat in excess of the true strength ; and in such 
cases Barlow's theory gives results more nearly correct. 

But if, instead of R, we use T or 6 y , whichever is smaller, in 
the formulas which we have deduced, and suppose that the 
neutral axis remains at the centre of the beam, we shall always 
be on the safe side / but there would often be an excess of 
strength, as, for instance, in the case of cast-iron the actual 
strength of the beam would be about twice as strong as that 
found by such a computation. 

The difficulty is avoided, practically, by using such a small 
fractional part of R as that it will be considered perfectly safe. 
This fraction is called the coefficient of safety. The values 



TKANSVEKSE STRENGTH. 157 

commonly used for beams are the same as for bars, and arc 
given in Article 43. 

Experiments should be made npon the material to be used in 
a structure, in order to determine its strength ; but in the ab- 
sence of such experiments the following mean values of It are 
used : — 

850 to 1,200 lbs. for wood, 
10,000 to 15,000 lbs. for wrought-iron, and 
6,000 to 8,000 lbs. for cast-iron. 



130. PRACTICAL FORMULAS, 

If B — 1,000 for wood, and 

12,000 for wrought-iron, 

we have for a rectangular oeam, supported at its ends and 
loaded at the middle of its length, 

P = — j — for wooden beams ; and 

_ 8000 M 2 
Jtr = j for wrought-iron beams. 

The length of the beam, and the load it is to sustain, are 
generally known quantities, and the breadth and depth are 
required ; but it is necessary to assume one of the latter, or 
assign a relation between them. For instance, if the depth 
be n times the breadth, the preceding formulas give 

} =V / eS? ;aildrf= V / w farwood (156) 

3/ ~pl 3 /~Pln 

and h = V ioo5? ; aBd d = V sow for wrou s ht - iron ; ( 15T > 

131. THE RELATIVE STRENGTH OF A BEAM Under 

the various conditions that it is supported or held is as the mo- 
ment of the applied forces ; hence, all the cases which have 
been considered may, relatively, be reduced to one, by finding 
how much a beam will carry which is fixed at one end and 



158 THE RESISTANCE OF MATERIALS. 

loaded at the free end, Equation (146), and multiplying the 
results by the following factors : — 

PACTOKS. 

Beam fixed at one end and loaded at the other 1 

" " " " uniformly loaded 2 

Beam supported at its ends and loaded at the middle 4 

" " " " uniformly loaded 8 

Beam fixed at one end and supported at the other, and 
uniformly loaded 8 

Beam fixed at both ends and loaded at the middle 8 

" " " " uniformly loaded 12 

If it is required to know the breadth of a beam which will 
sustain a given load, find b, from Equation (146) ; and for a 
beam in any other condition, divide by the factors given above 
for the corresponding case. 

If the depth is required, find d, from Equation (146), and 
divide the result for the particular case desired by the square 
root of the above factors. 



1 3&. EXAMPLES. 

1. A beam, whose depth is 8 inches, and length 8 feet, is supported at its 
ends, and required to sustain 500 pounds per foot of its length ; required its 
breadth so that it will have a factor of safety of -fa, B being 14,000 pounds. 

From Equation (146) we have, 

% 6PI 6x500x8x8x12 _„. . _ 
h = m> = 1400-x^ =^mches; 

and by examining the above table of factors we see that this must be divided 
by 8 ; . •. Ans. 3fV inches. 

2. If I = 10 feet, P at the middle = 2,000 lbs. ,5 = 4 inches, B = 1,000 lbs. , 
required d. Ans. 9.48 inches. 

3. If a beam, whose length is 8 feet, breadth is 3 inches, and depth 6 inches, 
is supported at its ends, and is broken by a weight of 10,000 pounds placed at 
the middle, and the weight of a cubic foot of the beam is 50 pounds ; required 
the value of It. Use Equation (150). 

4. If B = 80,000 lbs., I = 12 feet, b = 2 inches, d = 5 inches, how much 
will the beam sustain if supported at its ends and loaded uniformly over its 
whole length, coefficient of safety £? Ans. W = 9,259 lbs. 

5. A wooden beam, whose length is 12 feet, is supported at its ends ; re 
quired its breadth and depth so that it shall sustain one ton, uniformly distri- 



TRANSVEKSE STRENGTH. 159 

tmted over its whole length. Let R = 15,000 lbs., coefficient of safety -jfo and 
depth == 4 times the breadth. Ans. b = 2. 08 inches 

d = 8.32 inches. 

6. A beam is 2 inches wide and 8 inches deep, how much more will it sustain 
with its broad side vertical, than with it horizontal ? 

7. A wrought-iron beam 12 feet long, 2 inches wide, 4 inches deep, is sup- 
ported at its ends. The material weighs \ lb. per cubic inch ; how much load 
will it sustain uniformly distributed over its whole length, R = 54,000 lbs. ? 

Ans. Without the weight of the beam, 15,712 lbs. 

8. A beam is fixed at one end ; I = 20 feet, b — H inch, R = 40,000 lbs. ; 
weight of a cubic inch of the beam \ lb. Required the depth that it may sus- 
tain its own weight and 500 lbs. at the free end. Ans. 4.05 inches. 

9. The breadth of a beam is 3 inches, depth 8 inches, weight of a cubic foot 
of the beam 50 pounds, R = 12,000 ; required the length so that the beam 
shall break from its own weight when supported at its ends. 

Ans. I = 175.27 feet. 

133. RELATION BETWEEN STRAIN AND DEFLECTION. 

— When the strain is within the elastic limit we may easily find 
the greatest strain on the fibres corresponding to a given de- 
flection. For instance, take a rectangular beam, supported at 
its ends and loaded at the middle of its length, and we have 
from Equation (148) 

and from Equations (73) and (51) 

PI 3 
A — \ -™jb» which becomes, by substituting P from the pre- 
ceding, 

Ed 



* = l 



,'.P= 6 -^J (158) 

Examples. — 1. If I = 6 feet, 6 = 1\ inch, d = 4 inches, coefficient of elas- 
ticity = 25.000,000 lbs. is supported at its ends and loaded at the middle so aa 
to produce a deflection at the middle of A = f inch ; required the greatest 
strain on the fibres. Also required the load. 

2. On the same beam, if the greatest strain is R = 12,000 lbs., required the 
greatest deflection. 

3. If the beam is uniformly loaded, required the relation between the 
greatest strain and the greatest deflection. 



160 THE RESISTANCE OF MATERIALS. 

1 34. hollow rectangular reams. — If a rectangulai 
beam has a rectangular hollow, both symmetrically placed in 
reference to the neutral axis, as in Fig. 62, we may 
find its strength by deducting from the strength of 
a solid rectangular beam the strength of a solid 
beam of the same size as the hollow. But in this 
case, when the beam ruptures at b, the strain at b' 
will be less than B. As the strains increase di- 



b' 
d 



Fig. 62. 

rectly as the distance of the fibres from the neutral 

axis, we have, if d and d' are the depth of the outside and 

hollow parts respectively, 

d' 

\d : \d' : : B : strain at V = B —r. 

If b' = the breadth of the hollow, the stress on that part, if it 
were solid, would be, according to Equation (145), 



6 



(>&*-***£ 



which, taken from Equation (145), gives for the resistance of a 
hollow rectangular beam, 

ATi bd?-Vd' 5 ,^ x 

i*t — 2 — (W) 



If the hollow be on the outside, as in Fig. 63, 
forming an EL section, the result is the same. 



b 

Fig. 



135. IF THE UPPER AND LOWER FLANGES ARE UN- 
EQUAL it forms a double T, as in Fig. 64. Let the notation 
be as in the figure, and also d t equal the distance from the neu- 
tral axis to the upper element, and x the distance from the 
neutral axis to the lower element. 

To find the position of the neutral axis, make the statical 
moments of the surface above it equal to those below it. This 
gives 



TRANSVERSE STRENGTH. 



16 I 



a> v i&-\ $*) + ¥>'" {di - dj = a," w (x - i ar) + i y 

(x -d'J (160) 

We also have o\ = d - x = <Z' + tf" + <*'" - ». . (161) 
These equations will give x and d\. 




Fig. 64. 

Constructing the wedges as before, and the resistance to com- 
pression is represented by the wedge whose base is V o\ and 
altitude i?, minus the wedge whose base is (b ; — b f,f ) {d x — d') 

and altitude — -? — It. Hence the resistance to compression is 

\RVdi - i ^~ E (V - V") (a\ - d f ) 

The centre of gravity is at f the altitude, or %d\ for the for- 
mer wedge, and f (^ — d') for the latter, and if the volumes be 
multiplied by these quantities respectively, it will give for the 
moment of resistance to compression 

\RVd? - i | (V - &'") (a\ - d'f 
11 



162 THE RESISTANCE OF MATERIALS. 

Next consider the resistance to tension. Since the strains on 
the elements are proportional to their distances from the neu- 
tral axis, therefore 

7? 

d x \ x : : i? : strain at the lower side of the section = -r- x, 

and similarly, 

d x : (x — d f ') : : R : strain at the opposite side of the lower 

r> 

flange = -j- (x — d"). 

Hence the tensive strains will be represented by a wedge whose 

base is ~b"x and altitude -7- x, minus a wedge whose base is 

(5" _ £"/) (% _ #r} an( j altitude - T (x — d"). Hence the mo 
ment of resistance is 

i ^ V a? - i 2- {V - V") {x - d"f 

The total moment of resistance is the sum of the two moments, or 

i -j- [&' d? - {b f - V") (d, - dj + I" tf - {V - V") 



(x - d"f\ 



.(162) 



For a single T make b" and d" = in the above expression. 

The method which has here been applied to rectangular 
beams may be applied to beams of any form ; but it often re- 
quires a knowledge of higher mathematics to find the volume 
of the wedge, and the position of its centre of gravity ; or resort 
must be had to ingenious methods in connection with actual 
wedges of similar dimensions. 

130. TRUE VALUE OF d x AND AN EXAMPLE. In this 

and similar expressions 

d x = the distance from the neutral axis to the fibre most re- 
mote from it ON THE SIDE WHICH FIRST RUPTURES. 

.dx is usually taken as the distance to the most remote fibre, 






TRANSVERSE STRENGTH. 163 

without considering whether rupture will take place on that 
side or not ; but this oversight may lead to large errors. 

For example, let the dimensions of a cast-iron double T-beam 
be as in Fig. 65, and 228 inches between the 
supports. Kequired the load at the middle •, 3/4 

necessary to break it. ' 4 

The position of the neutral axis is found from 
Equations (160) and (161) to be 7.96 inches from 
the lower side, and 11.54: inches from the upper. 
As cast-iron will resist from four to six times as 
mueh to compression as to tension — this beam i/i\ ' ^_ 
will rupture on the lower side first ; hence d x in Fm 6 ^ 
the equation = 7.96 inches. As the value of P 
is not known, take a mean value = 36,000 lbs. The moment 
of the rupturing force — neglecting the weight of the beam — is 
J PI, which placed equal to Expression (162) and reduced gives 

p = Ms x ^r x - 1 ' 672 = 132 > 000 lbs - = 58 - 9 tons g ross - 

Had we used d x = 11.54, it would have given P = 40.0 tons. 
Such beams actually broke with from 50 to 54 tons ; or, in- 
cluding the weight of the beam, with a mean value of 52^- tons. 

By reversing the problem, and using 52-J- tons for P, we find 
that P is a little more than 32,000 pounds. Had this value of 
P been used in the first solution, and d 1 made equal 11.54, it 
would have given for P a little more than 36 tons, which would 
be the strength if the beam were inverted. If the upper flange 
were smaller or the lower larger, the discrepancy would have 
been greater. 

The strain upon a fibre in the upper surface is to the strain 
upon one in the lower surface as d x to x ; hence, if the material 
resists more to compression than to tension (as cast-iron), it 
should be so placed that the small flange shall resist the former, 
and the large one the latter. If a cast-iron beam be sup- 
ported at its ends, the smaller flange should be uppermost, and 
as it resists from four to six times as much compression as ten- 
sion, the neutral axis should be from four to six times as far 
from the upper surface as from the lower, for economy. Using 
the same notation as in Fig. 64, and we have, 



164 THE RESISTANCE OF MATERIALS. 

dx _ greatest compressive strain 
x greatest tensive strain 

and for economy we should have, 

d x _ ultimate compressive strength 
x ~ ultimate tensile strength 

The ultimate resistance of wrought-iron is greater for tension 
than for compresaion ; hence, if a wrought-iron beam is sup- 
ported at its ends, the heavier flange should be uppermost. 

The proper thickness of the vertical web can be determined 
only by experiment, and this has been done, in a measure, by 
Baron von Weber, in his experiments ou permanent way. 

137. experiments of baron von weber for deter- 
mining the thickness required for the central web of rails. 

Baron von Weber desired to ascertain what was the mini- 
mum thickness which could be given to the web of a rail, in 
order that the latter might still possess a greater power of re- 
sistance to lateral forces than the fastenings by which it was 




Fig. 65a. 

secured to the sleepers. For this purpose a piece of rail 6 feet 
in length, rolled, of the best iron at the Laurahutte, in Silesia, 
was supported at distances of 35.43 in., and loaded nearly to 
the limit of elasticity (which had been determined previously by 
experiments on other pieces of the same rail), and the deflec- 
tions were then measured with great care by an instrument 
capable of registering 10 * 0o in. with accuracy. This having 



TBANSVERSE STBENGTH. 



165 



been done, the web of the piece of rail was planed down, and 
each time that the thickness had been reduced 3 millimetres 
the vertical deflection of the rail under the above load was 
again tested, and the rail was subjected to the following rough 
but practical experiments. The piece of rail was fastened tc 
twice as many fir sleepers by double the number of spikes 
which would be employed in practice, and a lateral pressure 
was then applied to the head of the rail by means of a lifting- 
jack, until the rail began to cant and the spikes were drawn* 
The same thing was then done by a sudden pull, the apparatus 
used being a long lever fastened to the top of the rail, as shown 
in Fig. 65 a. The lifting-jack and the lever were applied to 
the ends of the rail, and the web of the latter had, in each case, 




i 

Fig. 655. 



to resist the whole strain required for drawing out the spikes. 
The results of the experiments made to ascertain the resistance 
of the rail to vertical flexure with different thicknesses of web, 
and under a load of 5,000 lbs., were as follows : — 



166 



THE BESISTA^CE OF MATEEIALS. 



Thickness of web. Vertical deflection. 

In. In. 

15 millimetres = 0.59. 0.016 

12 " 0.47 0.016 

9 " 0.35 0.019 

6 " 0.24 0.0194 

3 " 0.12 0.022 

These results showed ample stiffness, even when the web was 
reduced in thickness to 0.12 in. To determine the power of 
resistance of the rail to lateral flexure, an impression of the sec- 
tion was taken in lead each time that the spikes were drawn. 

The forces applied in these experiments were very far greater 
than those occurring in practice, jet it was found that with the 
web 12, 9, and even 6 millimetres thick, no distortion took place, 
and only when the thickness of the web was reduced to 3 milli- 
metres (0.12 in.) was a slight permanent lateral deflection of the 




Fig. 65c. 



head caused just as the spikes gave way. The section shown in 
Fig. 655 had then been reduced to that shown in Fig. 65<?. 



TRANSVEKSE STRENGTH. 167 

Next, a rail, with the web reduced to 3 mill. (0.12 in.) in 
thickness, was placed in the line leading to a turn-table on the 
Western Railway of Saxony, where it has remained until the 
present time, 1870, receiving the shocks due to engines passing 
to and from the turn-table more than one hundred times daily. 

It follows from these experiments that the least thickness 
ever given to the webs of rails in practice is more than suffi- 
cient, and that if it were possible to roll webs £ in. thick, such 
webs would be amply strong, if it were not that there would 
be a chance of their being torn at the points where they are 
traversed by the fish-plate bolts. Baron von Weber concludes 
that webs f in. or -J in. thick are amply strong enough for rails 
of any ordinary height, and that, in fact, the webs should be 
made as thin as the process of rolling and as the provision of 
sufficient bearing for the fish-plate bolts will permit. 

138. another graphical method. — If elementary 
processes are to be used for determining the strength, the fol- 
lowing method possesses many advantages over the former. 

Since the strains vary directly as their dis- 
tance from the neutral axis, the triangle 
ABC (Fig. 66), in the rectangle BODE, 
represents the compressive strains if each 
element of the shaded part has a strain 
equal to JR, ; and its moment is R times the 
area multiplied by the distance of the cen- 
tre of gravity of the triangle from the neu- F m. 66. 
tral axis ; or, 

Bx(hxi of id) xf of id^-^Md 2 , 

and the moment of tensile resistance is the same, hence the 
total moment is double this, or \Rbd?, as found by the preceding 
process. 

139. IF A SQUARE BEAM HAVE ONE OF ITS DIAGONALS 

vertical (Fig. 67), the neutral axis will coincide with the 
ether diagonal. Take any element, as ah, and project it on a 
line cd, which passes through A and is parallel to BO, and draw 



B 


\ / 


c 
















w 






/\ 


















/ \ 




E 




J) 



168 



THE RESISTANCE OF MATERIALS. 



the lines Oc and Od, and note the points f and g where they 
intersect the line ah. If the element were at cd, the strain upon 
it would be B, multiplied by the area of cd, or simply B.cd; 
but because the strains are directly proportional to the distances 
of the elements from the neutral axis, the strain on ah is Bfg. 
Proceed in this way with all the elements and construct the 
shaded figure. The strains on the upper part of the figure 




Fig. 67 

ABG, which begin with zero at B G, and increase gradually to 
R, at A, will be equivalent to the strains on the shaded figure 
A 0, if the strain is equal to B on each unit of its surface. 
Hence the total strain on each half is the area of the shaded 
part A 0, multiplied by B, and the moment of the strain of 
each part is this product multiplied by the distance of the 
centre of the shaded part from the axis BG. 
By similar triangles we have 

Aaiah:: AB : BO, and 
cd = ah : fg : : AO : x : : AB : Ba or AB — Aa ; 
x being the distance of fg from 0. 
From these eliminate ah, and find 



af= i(ah -fg) = £ 



BG 

(ABf 



(Aa? 



hence the curve which bounds the shaded figure is a parabola 
which is tangent to AB, and whose axis is parallel to BG. 
Let d = one side of the square, then 



TRANSVERSE STRENGTH. 169 

i y / 2d = A0, and 

J ^2d = the widest part of the shaded figure. 

The area of a parabola is two-thirds the area of a circumscribed 
rectangle. 

Hence the area of A is 



%x%)/2dxi^2d = id\ 



12 

and the moment is 



d s 

id 2 xii/2d=-^-^, 

and the moment of both sides, multiplied by i?, is 

*e?s (163 > 

If 5 = d in Equation (145) and the result compared with the 
above, we find : — 

The strength of a square beam with its side vertical : strength 
of the same beam with one of its diagonals vertical ; • j/2 : 1 
or as 7 : 5 nearly. 

So that increased depth merely is not a sufficient guarantee 
of increased strength. The reason why the strength is dimin- 
ished when the diagonal is vertical, is because there is a very 
small area at the vertex where the strain is greatest, but when 
a side is horizontal the whole width resists the maximum strain. 



140. irregular sections. — This method is applicable 
to irregular sections, as shown by the following example. 

Let Fig. 68 be a cross section of a beam. In a practical case 
it may be well to make an exact pattern of the cross section, of 
stiff paper or of a thin board of uniform thickness. To find 
the position of the neutral axis, draw a line on the pattern 
which shall be perpendicular to the direction of the forces 
which act upon the beam, that is, if the forces are vertical the 
line will be horizontal. In a form like Fig. 68, this line will 
naturally be parallel te the base of the figure. Then balance 
the pattern on a knife-edge; keeping the base of the figure (or 
the line previously drawn) parallel to the knife-edge, and when 



170 



THE RESISTANCE OF MATERIALS. 



it is balanced the line of support will be the neutral axis. Pro- 
ceed to construct the shaded part as shown in the figure, by pro- 




Fig. 68. 

jeeting any element, as ab on the line cd, and drawing cO and 
dO, and noting the intersections f and g, the same as in Fig. 67. 
The elements on the lower side must be projected on a line m?i, 
which is at the same distance from the neutral axis as the most 
remote element on the upper side. The area of the shaded 
part above the neutral axis should equal that below, because 
the resistance to extension equals that for compression. The 
area of the shaded part may be found approximately by di- 
viding it into small rectangles of known size, and adding 
together the full rectangles and estimating the sum of the frac- 
tional parts. Or, the shaded part may be cut out and carefully 
weighed and balanced by a rectangle of the same material, after 
which the sides of the rectangle may be carefully measured 
and contents computed. The area of the rectangle would evi- 
dently equal the area of the irregular figure. 

The ordinate to the centre of gravity of each part may be 
determined by cutting out the shaded parts and balancing each 
of them separately on a knife-edge, as before explained, keep 
ing the knife-edge parallel to the neutral axis. The distanc= 
between the line of support and the neutral axis will be the 
ordinate to the centre of gravity. The moment of resistance 
is then found by multiplying the area of each shaded jpart by 



TRANSVERSE STRENGTH. 



171 



the distance of its centre of gravity from the neutral axis, and 
multiplying the sum of the products by R. 

These mechanical methods may be managed by persons who 
have only a very limited knowledge of mathematics, and if 
skilfully and carefully done will give satisfactory results. It 
does not, however, furnish such an uniform, direct, and exact 
mode of solution as the analytical method which is hereafter 
explained. 



141. FORMULA OF STRENGTH ACCORDING TO BAR- 
LOW'S THEORY. — Either of the above methods may be used. One part 
of the expression for the strength i? of the same form as that found by the com- 
mon theory ; but instead of R we must use T, or G — the former if it ruptures 
by tension, the latter if by crushing. The other resistance, 0, for solid beama 
is evenly distributed over the surface. For example, take a rectangular beam, 
Fig. 61, and the resistance to longitudinal shearing on the upper side is <p b x 
{d=j <p bd, and its moment is \ <p bdx % of id = i <p bd 2 , and for both sides, 
£ <p bd 2 . Hence, according to Barlow's theory, the expression for the strength 
of a rectangular beam is 

[i <P + 6^1 bd 2 for cast-iron, and 



[i <p + £ C] bd 2 for wrought-iron and wood . 



.(164) 



If the beam is supported at its ends and loaded at the middle, we have 
iPl = [i + \T] bd 2 for cast-iron (165) 

The volume which represents the resistance due to is always a prism, hav- 
ing for its base the surface of the figure and 0, or some fraction of 0, for ita 
altitude. If the second method of illustration be used, it will take two figurea 
to fully illustrate the strains. For instance, if the section be as in Fig. 68, the 
moment of the shaded part will be multiplied by T or C, as the case may be. 
To find the remaining part of the moment, 
find the area of each part of the transverse 
section, also the distance of the centre of 
gravity of each part from the neutral axis. 
Then, to find the moment of resistance due to 
longitudinal shearing, multiply the area of each 
part by the distance of its centre of gravity 
from the neutral axis, add the products and 
multiply the sum by 0. This is true for solid 
sections ; but for hollow beams, T and J7 sec- 
tions, where there is an abrupt angular change 
from the flange to the vertical part of the 
beam, the factor requires a modification. 
For instance, take the simple case of a single T, Fig. 69, in which the breadth 
of the Tis b' and its depth d ', and the other notation as in the figure. 

The resistance of the upper part is represented by the prism whose base is 



d 




172 THE KESISTANCE OF MATERIALS. 

bx, and whose altitude is <j> plus the prism whose base is d' {b'~ 5), and whose 
altitude is — (j>. The resistance of the lower part is <pbd r . The total moment 

X 

of this resistance is — 

dJ 

<p bx.\x + d'(b'—b) x — 9 (x— \d') + $bd x x \d x . 
x 

To this add the moment of resistance for direct extension and compression, 
the expression for which is of the same form as for common theory, and we 
have for the total moment : — 

1^2 + —0(5'— 5) (x-\d) + \<p bd, 2 + £- [bd, z + b'x' 3 - (b' - b) 
(x - d'Y] (166) 

From numerous experiments made upon cast-iron beams having a variety 
of cross sections, Barlow found that <p varied nearly as T, that practically it 
was a fraction of T, the mean value of which was 0.9 T. 

For wrought-iron he found $ = 0.53 2* 

= 0. 6 G nearly. 

Peter Barlow, F.R.S., father of W. H. Barlow, F.R.S., the latter of whom 
proposed the "theory of flexure," in an article in the Civ. Eng. Jour., VoL 
xxi., p. 113, assumes that <f> = T. 

From the above it is inferred that the practical mean values of <p are : — 

16,000 lbs. for cast-iron. 
30,000 lbs. for wrought-iron. 
8,000 lbs. for wood. 

Example. — How much will a beam whose length is 12 feet, breadth 2 inches, 
depth 5 inches, sustain, if supported at its ends, and uniformly loaded over its 
whole length, and C = 50,000 lbs., <p = 30,000 lbs., and coefficient of safety i? 

Ans. — 11,000 lbs. nearly. 



142. BEAMS L.OABEB AT ANY NUMBER OF POINTS. — 

If the beam is loaded otherwise than has heretofore been sup- 
posed, it is only necessary to find the moment of all the forces 
in reference to the centre of a section and place the algebraic 
sum equal to the moments of resistance. Those which act in 
opposite directions will have contrary signs. 

For instance, if a beam, AB, Fig. 70, rests upon two sup- 
ports, and has weights P l5 P 2 , P 8 , etc., resting upon it at dis- 
tances respectively of n x , n^, %, etc., from one support, and 
mu ra 2 , m 3 , etc., from the other, the sum of the moments of the 
forces on any section G whose distance is x from the support 
.4, is 

Vi% — Px{% — %) — Pi(® — ih) — etc., 



TKANSVEKSE STBENGTH. 



173 



to include all the terms of P in which n is less than x. This 
equals % Bhd? for rectangular beams. 

X r 2 



q. . 


..::::z . n. 




■~n;~ 


P, 


[ p < IvJ 


P* 

<_ — ji 


J" — -^ 


— ari— I 


<K~~ 




c 1 
1 


— j 

.j 









Fig. 70. 

FJ, the reaction of one support, is readily found by taking 
the moments of all the external forces about B, and solving for 
Fi, thus :— 

V 1 l= Pim-L+Ptfriz+Pzmz + QtG., = 2 Pm 

ZPm 



V, 



Similarly V 2 



I 

2Pn 
I 



also, V x + V 2 = P x + P 2 + P 3 + etc., = 2 P. 



143. a partial uniform load. — Let the beam be 
'oaded uniformly over any portion 
of its length, as in Fig. 71. v tf? 



Let I = AB — length of beam ; 
2a = DE — length of the 
uniform load ; 



*I 



x = AFz=l the distance to 

any section ; 
w = the load on a unit of length ; 
V = the reaction of the support A ; 
G the centre of the load ; 
l 1 = AC; k= OB. 
Then AB =l L — a, and BF= x — I^ + a. 

Load on DF= w (a? — ^ -f a), 
" « D£ = 2wa. 



F 

Fig. 71. 




174 THE RESISTANCE OF MATERIALS. 

By the principle of moments 

o 
The moment of stress at i^is 

or ^^x~- iw (X-1. + a) 2 (167) 

That value of x which will make Equation (167) a maxim im, 
gives the position of the dangerous section. Differentiate, 
place equal zero, and make \ -f l 2 = l 9 and solve for x, and find 

x = a{l-^+l l (168) 

If \ = ±l,x = l 1 ; 
\ < \l, x > \ ; 
\ > il, x < l x ; 

so that the maximum strain is at the centre of the loading only 
when the centre of the loading is over the centre of the beam ; 
and in all other cases it is nearer the centre of the beam than 
the centre of the loading is. 

The maximum strain is found by substituting the value of a? 
Equation (168) in Equation (167). 

The following interesting facts are also proved. 

Let AD — y.\a — \ — y which in Equation (168) reduces 
it to 



h=(h-y)(l-^) (168a) 



which is a maximum for y = ; hence so far as AD is con- 
cerned, Equation (168a) is a maximum when one end of the 
load is over the support, and for this case the equation becomes 



-*-*-$) 



which is a maximum for \ — \ I or 2^ =%l, or the load must 
extend to the middle of the beam. Making a = ^ = i I, and 
Equation (168) becomes 



TRANSVERSE STRENGTH. 175 

and these values of \ and x in Equation (167) give for the max- 
imum moment of stress, 

rhw p = ^Wl (169) 

in which Wis the load on half the beam. 

Equation (167) gives the stress at the middle of the load, by 
making a = \ = J I and x = J I. This gives -J- Wl for the stress 
at the middle of the loading ; hence, the maximum stress is 1-J 
times the stress at the middle of the loading when the load 
extends from the one support to the middle of the beam. 

144. general formula. — The preceding methods are 
easily understood, and are perhaps sufficient for the more sim- 
ple cases ; but for the purposes of analysis a general formula is 
better, by means of which a direct analytical solution may be 
made for special cases. 

Let B, = the modulus of rupture, as explained in Article 129 , 

x and u horizontal coordinate axes, the former coinciding 
w T ith the axis of the beam, and y a vertical axis ; 

Then Rdudy = the resistance of a fibre which is most re- 
mote from the neutral axis ; 

Let <#! = distance between the neutral axis and the most re- 
mote fibre ; then, according to the common theory, 
since the strains vary as the distance from the neu- 
tral axis 

d 1 :y :: Rdudy : resistance of any fibre = —- ydydu 

.'. --- y 2 dydic = the moment of resistance of any fibre, 
and the sum of all the moments of resistance of any section is 

B r 



iff 



fdydu^ 



which is called the moment of rupture, and must equal the 
sum of the moments of the straining forces ; 

/. SPx =^I (170) 



176 THE RESISTANCE OF MATERIALS 

The second member of this equation involves the character < 

the material (i?) and the form of the transverse sections (-^-) ; tL6 

latter of which may. be determined by analysis, and the formei 
by experiment. The second member shows that for economy 
the material should be removed as much as possible from the 
neutral axis. 

Let R' = the strain on a unit of fibres at a distance d^ from 
the neutral axis, then 

SPx = f-7 (170a). 

By comparing Equations (170a) and (49) we see that 

~=^- - (171) 

which is true so long as the strain iZ'does not exceed the elastic 
limit. 



145. let the beam be RECTANGULAR, b the breadth, 
and d the depth, as in Fig. 61, 

+ id 
Then 7=4 / / y*dydu = 1 ^bd i 



Jo Jo 



d^ — ^d 

/. -r-I= \Rbd? which is the same as expression (145). 
d\ 



146. IF THE SIDES OF THE BEAM ARE INCLINED to the 

direction of the force, as in Fig. 72, let i be the inclination of 
the side to the horizontal ; then 

/= ^bd (dhmH + J 2 cos 2 *)* 
di = idsini -f- -J5cos i 

,.B±= \Rbd py + f c0 ^ 1 (172). 

d x L asim + ecosa J ' 

* See Appendix II. 



TRANSVERSE STRENGTH. 



177 



This expression has an algebraic minimum,* but not an alge- 
braic maximum. By inspection, however, we find that the 
practical maximum is found by making i = 90°, if d exceeds b. 
Hence, a rectangular beam is strongest when its broad side is 
parallel to the direction of the applied forces. 

Hence, the braces between joists in flooring, as in Fig. 73, 
not only serve to transmit the stresses from one to another, but 
also to strengthen them by keeping the sides vertical. 





Fig. 72. 



Fig. 73. 



If i = 90°, Equation (172) becomes \Rbd? (173). 

If I = d and i = 45°, Equation (172) reduces to 

*S < m > 

(which is the same as Expression (163) ), 
and if b = d, and i — 0° or 90°, it becomes 

\Bd\ 
Hence, the strength of a square beam having a side vertical 
is to the strength of the same beam having its diagonal verti- 
cal, as 

1: Vh 
or |/2 to 1 or as 7 to 5 nearly. 
In establishing Equation (172) it was assumed that the neu- 
tral surface was perpendicular to the direction of the applied 
forces, which is not strictly true unless the forces coincide with 
the diagonal ; for in other cases there is a stronger tendency to 



* See an article by the author in the Journal of Franklin Institute, VoL 
LXXV., p. 260. 




178 THE RESISTANCE OF MATERIALS. 

deflect sidewise than in the direction of the depth. In this case ; 
as soon as the beam is bent there is a tendency to torsion. Both 
these conditions make the beam weaker than when the sides 
are vertical. If the tendency to torsion be neglected, the case 
may be easily solved ; bnt the result shows the advantage of 
keeping the sides vertical. 

147. THE STRONGEST RECTANGULAR BEAM which Can 

2L^— -^ b be cut from a cylindrical one has the breadth 
to the depth as 1 to |/2, or nearly as 5 to 7. 

Let x — AB — the breadth, 
y = AC = the depth, and 
D = AD = the diameter. 



f = D 2 - a? 

and Expression (173) becomes 

^Rxtf = \Bx {D* - a?), 

which by the Differential Calculus is found to be a maximum 
for 

x = Z>i/i.'.y = DVi 
,\ x : y : : 1 : |/2 or nearly as 5 to 7. 

Examples. — How much stronger is a cylindrical beam than the strongest rec- 
tangular one which can be cut from it ? 

(For the strength of a cylindrical beam, see Equation (180) .) 

Ans. — About 53 per cent. 

How much stronger is the strongest rectangular beam that can be cut from 
a cylindrical one, than the greatest square beam which can be cut from it ? 

148. triangular beams. — If the base is perpendicular 
to the neutral axis, as in Fig. 75 ; 

Let d — AD = the altitude, and 
b = £0= the base. 

Take the origin of coordinates at the centre of gravity of the 
triangle, y vertical and %i horizontal. 
Then, by similar triangles. 



.-. I = fftfdydu 
"We also have 

I 



TRANSVERSE STRENGTH. 

lb:y\:d\$d-\-u 

idb + ¥>u> j %d j 

ydu = -fedb*. 



179 



-•/ 



.-. E ~ = ^i? <# 2 = ^i?^ i 



.(175) 



in which A is the area of the triangle. 




Fig. 75. 




If the base is parallel to the neutral axis, as in Fig. 76, then, 
by similar triaogles, 

d : §b : : id — y : u 

b 



* = (f*-y) 



U 



I I t 

J -Id 



id 
tfudy 
id 



I=2/ftfdydu = 2 






We also have 
d 1 = %d 



* This is more easily solved by taking the moment about an axis through 
the vertex and parallel to the base, and using the formula of reduction. See 
Appendix. 



180 



THE RESISTANCE OE MATERIALS. 



R 



I 



^Rbd 2 = -h 



RAd. 



(176). 



Expressions (173) and (175) show that a triangular beam 
which has the same area and depth as a rectangular one, is only 
half as strong as the rectangular one. 

Some authors have said that a triangular beam is twice as 
strong with its apex up as with it down, but this is not always 
the case. If the ultimate resistance of the material is the same 
for tension as for compression, the beam will be equally strong 
with the apex up or down. 

If the beam is made of cast-iron, and supported at its ends, it 
will be about 6 times as strong with the apex up as down ; but 
if the beam be fixed at one end, and loaded at the free end, it 
will be about 6 times as strong with the apex down as with it up. 

149. trapezoidal beam. — Required the strongest trap 
ezoidal beam which can be cut from a given triangular one* 

C Let ABO be the given triangle, 

/\ ABED the required trapezoid, 

d = CG = the longest altitude, 
b = AB, di = FR, w= OF, 
z = OR = d x + w, and v = DE. 

Uis the neutral axis of the trapezoid, 
which passes through its centre of gravity 
H. We may then find : — 




hv 



, - d W- 

d >=h x —b 



i d d r b 5 + Vv - 85V + 85 V 



■ v 1 - ih^T 



b 5 + b 4 v 



bv* — v 5 ' 

b + v~~ 

85V 4- 85V -bv* 



d\ 



2b 2 - bv 



] 

-J (177) 



which is to be a maximum. By the Calculus we find, after re- 



duction, that 



* See an article by the aafchar in the Journal of Franklin Institute, Vol 
tTJ, , third series, p. 198. 



TRANSVERSE STRENGTH. 181 

V Z + 5fo 2 + Wv -¥=0, 

for a maximum, which solved gives 

v = 0.130935 or 0.135 nearly, and hence 
w = 0.13093d or 0.13d (178) 

which substituted in (177) gives 

rI- = 0.545625 ^ (179). 

Dividing Equation (179) by Equation (176) gives 1.09125 ; hence 
from (178) and (179) we infer that if the angle of the prism be 
taken off 0.13 of its depth, the remaining trapezoidal beam will 
be 1.091 times as strong as the triangular one, which is a gain 
of over 9 per cent. 

In order to explain this paradox it must be granted that the 
condition does not require that the beam shall be broken in 
two, but that a fibre shall not be broken — in other words, the 
beam shall not be fractured. The greatest strain is at the edge, 
where there is but a single fibre to resist it ; but, after a small 
portion of the edge is removed, there are many fibres along the 
line DE, each of which will sustain an equal part of the 
greatest strain. 

If the triangular beam were loaded so as to just commence 
fracturing at the edge, the load might be increased 9 per cent, 
and increase the fracture to only thirteen-hundredths of the 
depth ; but if the load be increased 10 per cent, it will break 
the beam in two. 

These results are independent of the material of which the 
beam is made. If the beam be cut off \ the depth, its strength 
is found from Equation (177) to be 

Ebd? 



0.465608 



12 



which is 0.93101 of Equation (176). 

Mr. Couch found* for the mean of seven experiments on tri- 
angular oak beams of equal length, that they broke with 306 
pounds. The mean of two experiments on trapezoidal oak 

* See Barlow's Strength of Materials. 



182 THE RESISTANCE OF MATERIALS. 

beams, made from triangular beams of the same size as in the 
preceding experiments, by cutting off the edge one-third the 
depth when the narrow base was upward, was 284.5 pounds. 
This differs by less than half a pound of 0.931 times 306 
pounds. 



150. cylindrical beams. — The moment of inertia of a 
circular section in which r is the radius, is 

r 

d 1 =r; 

/.^=i^7r^ (180) 

If polar coordinates are used, we have 
dudy == pdpd<j) y 
where p is a variable radius and <£ a variable angle. 
Also y = p sin <p 

.\I==jyfdydu = / / phm % <f>dpd$ 

Jo Jo 

= i^y i(l — cos 2< ft # = i 73 ^ 4 ? as before « 
For a circular annulus we have 

By comparing Equations (180) and (145) we see that the 
strength of a cylindrical beam is to that of a circumscribed 

rectangular one as ^ : -J-, or as 0.589 + : J. 

Also the strength of a cylindrical beam is to that of a square 
one of the same area as \RAd' to \BAd (d! being the diameter 
of the circle), 




TKANSVEKSE STRENGTH. 



183 



, or as 1 : (f-^ = f 4/77- j or as 1 : 1.18 nearly. 

It may be shown in the same sense as explained in the pre- 
ceding article, that if a thin segment be removed from the 
upper and lower sides of the beam it will be stronger. 



151. ELLIPTICAL BEAMS. 

Let b — the conjugate axis, and 
d = the transverse axis ; then 
if d is vertical (Fig. 79), we have 
I—^z nbd* and d x = \d. 

If b is vertical (Fig. 80), we have 

i"= $p:b*d and d x = \b. 





152. PARABOLIC BEAMS. 





If b = the base, and 

d = the height of the parabola, and 
if d is vertical (Fig. 81), we have 

I = j^gbd 3 , and d x = fd. 

If b is vertical (Fig. 82), then 

I = ■g^W, and d x = }b. 



184 



THE KESISTANCE OF MATERIALS. 



153. ACCORDING TO BARLOW'S THEORY we have 

^^ffy 2 dydy^^^ffydydu = ^Px (181) 

which must be integrated between the proper limits to include the whole 
section. 

If the neutral axis is at the centre of the sections, and the beam is rectan- 
gular, we have 

T r /» 6 r+ id 1 n+ id /* b 

ST^W L/o/f^H + Vo /o *** = SPw ' 

which reduced gives 

%Tbd 2 + {<pbd 2 = tV [2T+ty]bd* ; 

hence, if <p has any ratio to T 7 , the law of resistance in solid rectangular beams 
is the same as for the common theory only. 

If = T, this becomes 

&Tbd 2 . 



154. oblique strains.— If the force be inclined to the 
axis, as in Fig. 83, let 6 = the angle which P makes with the 
axis of the beam. 




Fig. 83. 

Then P 2 = P cos 6 = longitudinal component, 
P x = P sin 6 = normal component. 

If K= the transverse section, then 

P cos 6 ,, , . .. . 

= — = the tension or compression upon a unit of 

section due to the direct pull or push. This tends directly to 
diminish the tabular value of B, in the formula. If the beam 
be fixed at one end and P be applied at the free end, as in 
Fig. 83, the equation cf moments becomes 



Px sin 6 



-(*-*?-•) 



which for rectangular beams becomes 



TRANSVERSE STRENGTH. 185 

Pa; sin 6 = ^ iTylT ^ * 

in which 6 is always acute. 

This solution does not recognize any deflection. If the di- 
rection of P % does not intersect the neutral axis at the fixed end 
it will have a moment. 

If flexure is considered, we find the strain upon the most 
remote fibre from the neutral axis at the fixed end, to which 
add the strain due to a direct pull (or push), which sum should 
not exceed the tabular value of R. 

From Equation (171) we have 

£' = —% 

P 
which is the strain on a unit of the extreme fibres. 
From Equation (130) we have 

= E{-fl + q*A) 

at the fixed end where the strain is evidently a maximum. 
From the Equations following (130) we find 

which substituted above gives 

- Rr _ d x Ef[f l -e-< 1 ) 
q (e ql + e-Q*) 

_ d x P sin (f l — e~ ql ) 
Iq (e ql + <T**) 



K^ Iq \e* + e-« l ) 

. P - V IE (a P * \ ( eQl + ** \ 
1_ ck \ KB) \e« l - e-« l J 



In the solution thus far we have supposed that rupture takes 
place on the side of tension, but if it should take place on the 
compressed side, we would have 



186 



THE RESISTANCE OF MATERIALS. 






^ KB) \e« - <r«) 



The total load, P, which the beam can sustain in these cases 
can be found only by a series of approximations, since P % and 
q both involve P. 

The solution of the case shown in Fig. 83a, when flexure is not 




MM 
Fig. 83a. 




Fig. 835. 



considered, is given in " Bridges and Roofs," p. 20. If flexure 
is considered, the reactions at the ends will be treated as the 
oblique forces, and the solution made substantially as in the 
preceding case. 



155. POSITION OF THE NEUTRAL AXIS FOR MINIMUM 
STRENGTH. 

Let I = the moment of inertia of the section when the axis 
passes through the centre of gravity of the sec- 
tion, 
I = the moment of inertia of the same section about an 

axis parallel to the former, 
D = the distance between the axes, 
A — the area of the section, and 

#! = the ordinate to the most remote fibre from the 
centre, 
Then D -b a^ = the ordinate to the most remote fibre from the 
second axis, and 

i __ i + Arc 



d x ' Oi + D 
which is a minimum for 



.(181c) 



i>=4-i±*/i + 2y 



TEANSVEESE STEENGTH. 187 

One of the roots is positive and less than d^ and the other 
is negative and greater than d^ but both give an algebraic 
minimum. 

For a rectangle A = bd, I = -^bd s , and a x = \d. 

.\ D = 0.07732<Z or - 1.07732J. 

Using the positive value, we have 

J2^=0.1te7Md 2 : 
d x 

which is only 0.9282 of the strength when the axis passes through 
the centre. 

If the sections are circular 

D = 0.11807/* 
and 

~ = 0.74157^, 
d x 

which is 0.9441 of the strength when the axis passes through the 
centre. 

Has this analysis any physical signification ? Being entirely 
independent of the character of the material, it does not 
explain the difference between the values of R and T or O, 
So far as the analysis is concerned there is nothing to determine 
which way the neutral axis will move from the centre. 

In some cases, practically, we might have d^ = a x — D ; in 
which case we have for a minimum 



*=*[i±/i+23ij 



which for rectangular beams gives 

D = 1.077^ or - 0.0T7&. 



188 THE RESISTANCE OF MATERIALS. 



CHAPTER VII. 

BEAMS OF UNIFORM RESISTANCE. 

156. general expression. — If beams are so formed 
that they are equally liable to break at every transverse section, 
they are beams of uniform resistance, and are generally called 
beams of uniform strength. The former term is preferable, 
because it applies with equal force to all strains less than that 
which will produce rupture. In such a beam the strain on the 
fibre most remote from the neutral axis is uniform throughout 
the whole length of the beam. The analytical condition, ac- 
cording to the common theory, is : The sum of the moments of 
the resisting forces must vary directly as the sum of the moments 
of the applied forces ; hence Equation (171) is applicable ; or 

2Px = ^ (182) 

which must be true for all values of x. In addition to this 
the transverse shearing strain must be provided for. To obtain 
practical results it is necessary to consider 

PARTICULAR CASES. 

157. BEAMS FIXEJ* AT ONE END AND LOADED AT THE 

free end. — Required the form of a beam of uniform resist- 
ance when it is fixed at one end and loaded at the free end, 
1st. Let the sections be rectangular, and 

y = the variable depth, and 
u — the variable width. 

Then /= tV^/ 3 ( see Equation (51) ), 
d x = iy, and 
2 Px = Px = the variable load.* 

* For 2P# use the general moments as given in the table in Article 109, so 
far as they are applicable. 



BEAMS OF UNIFORM RESISTANCE. 189 

Hence Equation (182) becomes 

Px = \Rmf (183) 

a. Let the breadth be constant ; or u = b ; then (183) be- 
comes 

Px = \Bbtf . . .(184) 

which is the equation of a parabola, whose axis is horizontal 

t p 



and parameter is -^r-. See Fig. 84. 





Fig. 84. Fig. 85. 

b. Suppose that the depth is constant, or y = d. Then (183) 
becomes 

Px = \Rd*u (185) 

which is the equation of a straight line ; hence the beam is a 
wedge, as in Fig. 85. 

c. If the sections are rectangular and similar, then 

u : y : : b : d 
b 

and Equation (183) becomes 

which is the equation of a cubical parabola. 

2d. Let the sections be circular. Then 
1= -g^n/ 4 Equation (52), in which y is the 
diameter of the circle), and d l = \y ; hence 
(182) becomes 

which is also the equation of a cubical para- 
bola, as shown in Fig. 86. 
3d. Let the transverse sections be rectangular, and /constant, 

the breadth and depth both being variable, then Equation (182) 

becomes 




190 



THE RESISTANCE OF MATERIALS. 



p x = R^l = R± (186) 

in which e is a constant, == bd 5 , b and d being the breadth and 
depth at the fixed end. Equation (186) is the equation of the 
vertical longitudinal sections, and is the equation of an hyper- 





Fig. 87. 



Fig. 88. 



bola referred to its asymptotes. See Fig. 87. If the value of 
y from this equation be substituted in the Equation uy* = c, it 
gives 

U = ~RW (187) 

which is the equation of the horizontal longitudinal sections ; 

hence they are cubical parabolas, as in Fig. 88. For % and u = 

n ^* 7 * 216P 3 Z 3 , 6 PI 

0,^oo,andfor*=^ = J =1 ^^ = _ 

4th. If the breadth is the nth. power of the depth, and the 
sections are rectangular, then u = y n , and Equation (183) 
becomes 

Px = %Rutf = \Ry n +\ 

which is the general equation of parabolas. 



158, BEAMS FIXED AT ONE END AND UNIFORMLY 

IjOjlded. — Required the form of a beam of uniform resistance 



BEAMS OF UNIFORM RESISTANCE. 



191 



when it is fixed at one end and uniformly loaded over its whole 
length ; the weight of the beam being neglected. 

The origin of coordinates being still at the free end, we have 
wx = the load on a length x, and 
^wx 2 = the moment of the load (Equation (53) ). 
Hence, for rectangular sections, Equation (182) becomes 

iwx* = \Rutf (188) 

a. If the breadth is constant, or u = b in (188), it becomes 
%wa? = ^Rby*) 
which is the equation of a straight line , and hence the beam 
will be a wedge, as in Fig. 89. 







WiiiiliiM 


1 1 1 


1 1 


1 1 1, 






1 i 1 




1 ' \ 




\ 1 1 



Fig. 89. 



b. Let the depth be constant ; or y = d in (188) 
/. fywx 2 = \Bd % u ; — 
a parabola whose axis is perpendicular to the axis of the beam, 
as in Fig. 90. 





Fig. 90. 
C. Let the sections be similar : — 



Fig. 91. 



then d : b : : y : u = -^y, 



192 THE RESISTANCE OF MATERIALS. 

,\ Equation (188) becomes %wa? = \R -y* ; — 

Cu 

a semi-cubical parabola, as in Fig. 91. 

d. Let / be constant, or -^ 2 u y z = tV^ 3 * Then Equation (182) 

becomes 

bd s 
\wv? = \R ; — an hyperbola of the second order. 

if 

159. previous cases combined. — Required the form 
of the beam of uniform resistance when it is fixed at one end 
and loaded uniformly, and also loaded at the free end. 

The moment of applied forces is Px+^wx 2 ; hence Equation 
(182) becomes, for rectangular beams, 

Px -f- ^wx 2 = ^Ruy 2 . 

Hence, if the depth is constant, Px + Jme 2 = \Rud 2 ; — a 
parabola. 

Hence, if the breadth is constant, Px + %wx 2 = ^Rby 2 ; — a 
hyperbola. 

Hence, if the sections are similar, Px + %wa? — %R -y* ;•— a 

Cu 

semi- cubical parabola. 

160. weight of the beam considered. — Required 

the form of the beam of uniform resistance when the weight 
of the beam is the only load ; the beam being fixed at one end, 
and free at the other. 

a. Let the sections be rectangular and the breadth constant. 
Let x *= AB ; Fig. 92, 
y = DC, 

b — the breadth, and 
8 =± the weight of a unit of volume. 

Then fydx = the area ADO, and 

8b fydx = the weight of ADC; 

the limits of integration being and x. 
If i^is the centre of gravity of ADO; we have, 



AF= 



fxydx 
fydx' 



BEAMS OF UNIFOEM STRENGTH. 



193 




Fig. 92. 




The moment of the applied forces is the weight of ADO 
multiplied by the distance BF= x — AK Hence, Equation 

(182) becomes 



»/**[— ^£]=w. 



which reduced gives 






(189) 



which is the equation of the common parabola, the axis being 
vertical. 

b. Let the depth be constant. In a similar way we find 



U 



This solved gives 

* = ~ \llr Na P- l0 s- [vlw ° + * ~ M ] + °'> 

in which Cand C are constants of integration, and involve 
the position of the origin of coordinates and direction of the 
curve at a known point. 

c. Let the beam be a conoid of revolution^ as in Fig. 93. 
We have, as before {y being the radius of the circle), 

which reduced gives 
13 



194 



THE RESISTANCE OF MATERIALS. 



* s 



.(191) 



which is the equation of the common parabola. 

d. Suppose, in the preceding cases, that an additional load, 
P, is applied at the free end. 

Some of the equations which result from this condition can- 
not be integrated in finite terms, and hence the curves cannot 
be classified. 



161. BEAMS SUPPORTED AT THEIR ENDS. 

A. Let the beam be supported at its ends and loaded at the 
middle point. 

For this case, Equation (182) becomes, for rectangular 
tions, 

iPx = \Buy % (192) 

a. If the breadth is constant, we have 
which is the equation of the common parabola. 





Fig. 94. 



Fig. 95. 



The beam consists of two parabolas, having their vertices, 
one at each support, as in Fig. 94. 
b. If the depth is constant, we have 

\Px = \Rd*u (193) 

a wedge, as in Fig. 95. 





Fig. 96. 



Fig. 97. 



B. If the beam is uniformly loaded, we have from Equations 
(74) and (182), 



BEAMS OF UNIFORM STRENGTH. 



195 



%w (Ix — x 2 ) = \B,uy 2 — if rectangular, and if the breadth is 

constant, %w (Ix — x 2 ) = \Bby 2 (194) 

an ellipse, Fig. 96. 

If the depth is constant, %w (Ix — x 2 ) = \B,d 2 u* a parabola, 
Fig. 97. 

C. Let the beam have an uniform load and also an uni- 
formly increasing load from one end to the other, as m Fig. 
98. 

v 





Fig. 98. Fig. 99. 

Let W= the weight of the uniform load, 

TTi = the weight of the uniformly increasing load, and 
V = the reaction of the support at the end which has the 
least load. 
Then V=bW+iWi. 
Let x be reckoned from A. then the load on x is 

and the moment of this reaction and load on a section which is 
at a distance x from A is 



{iW+iWJx 



Wx 2 

21 






(195) 



which equals \Rby 2 for rectangular beams of uniform breadth. 
To find the point of greatest strain, make the first differential 
coefficient of (195), equal to zero. We thus find 



iTr+iTT!-^- ] 



«? = 0. 



If W = 0, this gives 



x = il |/3. 



When W= 0, this becomes the case of a fluid pressing against 
a vertical surface. 



196 THE EESISTANCE OF MATERIALS. 

162. beams fixed at their ends. — If the beam is 
, fixed at its ends and loaded at the middle with a weight, P, we 
have, from Equations (117) and (182), when the breadth is 
uniform, 

iP(l - 4x) = \Bbf , (196) 

which is the equation of a parabola. The beam really consists 
of four double parabolas with their vertices tangent to each 
other, as in Fig. 99. The vertices are %l from the end. 

If the load were uniform we would obtain, in a similar way, 
a beam composed of four wedges. These are direct deductions 
from the common theory. 

This shows in a very marked degree the absurdity of not 
providing for the transverse shearing strain. All of the pre- 
ceding cases show the same absurdity. The section being 
reduced to naught leaves no ability to resist the shearing strain. 
In a case like Fig. 99, it even prevents the equation of moments 
from being practically realized ; for the resisting forces cannot 
be transmitted past the points A and B. 

163. EFFECT OF TRANSVERSE SHEARING STRESS 071 

modifying the forms of the beams of uniform resistance. 

Take, for example, the case of a beam supported at its ends 
and uniformly loaded. The transverse shearing strain is 

Ss = \wl — wx = iw(l — 2x), 

which is the equation of a straight line, Fig. 100. 





Fig. 100. Fig. 101. 

The double ordinate at the end is 

\wl -r(Jx modulus of shearing, #,) 

in which b is the breadth of the beam. 

If the resistance to transverse shearing varies directl) as the 
transverse section, then will the triangle A OB represent the 
vertical section of one-half of a beam of uniform strength when 



BEAMS OF UNIFORM RESISTANCE. 197 

shearing alone is considered. This result is as absurd as the 
preceding. 

Practically, the two cases may be combined by adding the 
ordinates of the line A O to those of the ellipse, the result being 
shown in Fig. 101. 

Theoretically, I do not see how they can be combined, since 
the conditions upon which the equations are established are not 
only independent, but are not simultaneous. Each condition 
furnishes a determinate equation* One is an equation of mo- 
ments, and the other of forces. The practical solution above 
suggested, doubtless gives an excess of strength at all points, 
except at the ends and middle ; for by increasing the depth we 
increase the moments of resistance, and probably add more 
than is necessary to resist the transverse shearing, since that is 
greatest near the neutral axis where the strain from moments 
is least. 



164. unsolved problems. — When the nature of the 
problem is such that it cannot be solved in accordance with the 
principles of ultimate strength, resort must be had to direct 
experiment, as has been the case with the Strength of Columns, 
The Collapse of Tubes, etc. ; or the solution must be effected 
in accordance with the principles of The Perfect Elasticity of 
Solids. Of the following problems the 1st, 2d, 3d, and 4th are 
considered as unsolved by the principles of ultimate strength.' 
For a solution of the 5 th one see Yan Kostrand's Eng. Mag. 
for March, 1877. For a solution of some of the others in accord- 
ance with the theory of elasticit} 7 , see Grashofs Festigheits- 
lehre, Berlin, 1866. 

1. Required a formula for the strength of a circular plate of 
iron of uniform thickness, supported throughout its circumfer- 
ence and loaded uniformly. 

* To illustrate, suppose it is required to find the radius of a sphere whese 
volume equals (numerically) the area of the surface ; and whose diameter 
equals (numerically) the area of the hemisphere. The former gives r = 3, 

and the latter, r = — • 



198 



THE RESISTANCE OF MATERIALS. 



2. Required the strength of the same plate if the edges are 
bolted down. 

3. Required the equation of the curve for each of the pre- 
ceding cases, that they may have the greatest strength with a 
given amount of material. 

4. In the preceding problems, suppose that the plate is 
square. 

5. Required the form of a beam of uniform strength which 
is supported at its ends, the weight of the beam being the only 
load. Suppose, also, that it is loaded at the middle. 



165. BEST FORM OF CAST-IRON BE] AM AS FOUND EX- 
PERIMENT a L.ii it. — Cast-iron beams were first successfully used 
for building purposes by Messrs. Boulton & Watt. The form 
„ of the cross-section of the beams which 



*M 



Fig. 102. 



JQ they used is shown in Fig. 102. More 
recent experiments show that this is a 
good form, but not the best. 

About 1822 Mr. Tredgold made an 
experiment upon a cast-iron beam of the 
form shown in Fig. 103, to determine its 
deflection. He recommended this form 
for beams. 

Mr. Fairbairn has justly the credit of 
l ~J/ 4 making the first series of experiments for 
determining the best form of the beam. 
' These experiments were prosecuted by 



himself for a few years, beginning about 1822, and continued 
still later by Mr. Hodgkinson. 

The expej ments quickly indicated that the lower flange 
shoild be considerably the largest. 



BEAMS OF UNIFORM RESISTANCE. 199 

The following experiments were made by Mr. Hodgkinson 
(Fairbairn on Cast and Wrought- Iron, p. 11). 

Fig. 104 shows the elevation and cross-section of a beam 
whose dimensions are as follows : — 




3 



Fig. 104. 



Area of top rib = 1.75 x 0.42 = 0.735 inches. 
Area of bottom rib = 1.77 X 0.39 = 0.690 " 

Thickness of vertical rib ... , 0.29 " 

Depth of the beam 5.125 " 

Distance between the supports 54.00 " 

Area of the whole section 2.82 square inches. 

Weight of the beam 36 J pounds. 

Breaking weight 6,678 pounds. 

The form of the fracture is shown at b n r. It broke by 
tension. 



Dimensions. 
Thickn 

a 


Experiment 

Inches. 
ess at A = 0.32 
" B = 0.44 
" (7 = 0.47 
" FJE = 2.27 
" DE= 0.52 
of the beam = 5.125 
f the section = 3.2 sqi 


IY, 


A, 


a 


2? X 




-^T^i 


Depth 
Area o 


Lare 


Fig. 105. 
inches. 



Distance between the supports = 54 inches. 
Weight of casting = 40£ pounds. 
Deflection with 5,758 pounds = 0.25 inches. 

« " 7,138 " = 0.37 " 

Breaking weight 8,270 pounds. 



200 



THE RESISTANCE OF MATERIALS. 



Experiment 19. 



t 



^L 



Fig. 106. 



Dimensions in inches : — 

Area of top rib = 2.33 x 0.31 = 0.72. 

Area of bottom rib = 6.67 X 0.66 = 4.4. 

Ratio of the area of the ribs = 6 to 1. 

Thickness of vertical part = 0.266. 

Area of section, 6.4. 

Depth of beam, 5J-. 

Distance between the supports, 54 inches. 

"Weight of beam, 71 pounds. 

This beam broke by compression at the middle of the length 
with 26,084 pounds. 

It is probable that the neutral axis was very near the vertex 
n, or about f the depth. 



Experiment 21. 




Fig. 107. 

This was an elliptical beam, Fig. 107. 

Dimensions in inches : — 

Area of top rib = 1.54 x 0.32 = 0.493. 

Area of bottom rib = 6.50 x 0.51 = 3.315. 

Eatio of ribs, 6£ to 1. 

Thickness of vertical part = 0.34. 

Depth of beam, 5 J. 

Area of the section, 5.41. 



ii 


ii 


ii 


ii 


a 


ii 



BEAMS OF UNIFOKM RESISTANCE. 201 

Distance between supports, 54 inches. 

Weight of beam, 70 j- pounds. 

Broke at the middle by tension with 21.009 pounds. 

Form of fracture b nr\ b n = 1.8 inches. 

As these beams have all the same depth and rested oh the 
same supports, 4 feet 6 inches apart, their relative strengths 
will be approximately as the breaking weight divided by the 
area of the cross section. 

In Experiment 1, 6,678 -> 2.82 == 2,368 lbs. per square inch. 
" " 14, 8,270^3.2 =2,584 

" " 19, 26,084 -f- 6.4 =4,075 

" " 21, 21,009 -^ 5.41 = 3,883 

It is evident from these experiments, that when the vertical 
rib is thin, the area of the lower rib should be about 6 times 
that of the upper. In the 19th experiment it has already been 
observed that the beam broke at the top, and in the 21st it 
broke at the bottom, although the lower flange was larger in 
proportion to the upper than in the preceding case, and the 
comparison shows that they were about equally well propor- 
tioned. They should be so proportioned that they are equally 
liable to break at the top and bottom. 

A beam proportioned so as to be similar to either of the two 
last forms above mentioned may be called a " type form." 



166. hodgkinson's formulas/w the strength of cast- 
iron beams of the type form. 

Let W = the breaking weight in tons (gross). 

a = the area of bottom rib at the middle of the beam. 

d = the depth of the beam at the middle, 
and I = the distance between the supports. 
Then according to Mr. Hodgkinson's experiments we have 

W= 26 — when the beam is cast with the bottom 



rib up, and 



W = 24 —when the beam is cast on its side. 

V 



THE KESISTANCE OF MATERIALS. 

167. experiments on T rails. — Experiments on T 
bars, supported at their ends and loaded at the middle, gave 
the following results : — * 

Hot blast bar, rib upward, X broke with 1.120 pounds. 

" " " downward, T broke with 364 " 

Cold blast bar, rib upward, J. broke with 2,352 " 

" " " downward, T broke with 980 " 

The ratio of the strengths is nearly as 3 to 1, but according 
to the table in Article 47, we might reasonably expect a higher 
ratio. If a greater number of experiments would not have 
given a higher ratio, we would account for the discrepancy by 
supposing that the neutral axis moved before rupture took 
place, or that the ratio of the crushing strength to the tenacity is 
less for comparatively thin castings than for thick ones. It is 
known that the crushing strength of thin castings is proportion- 
ately stronger than thick ones. Hodgkinson found that for 
castings 2, 2-J-, and 3 inches thick, the crushing strengths were 
as 1 to 0.780 to 0.756 ; and Colonel James found a greater in- 
crease — being as 1 to 0.794 to 0.624. See also Article 41. 

168. wrougpt-iron beams. — The treacherous character 
of cast-iron beams has led to the introduction of solid wrought- 
iron ones. Special machinery and special processes of manu- 
facture have been brought into use, by means of which they 
are quickly and cheaply made. They are usually of the double 
T (I) section. 

169. A NOVEL AND PECUIilARIiT CONSTRUCTED FLOOR 

is here given as an illustration of the use of a plate (see Arti- 
cle 164, No. 4).f It was executed in Amsterdam, for a floor 
60 feet square. The flooring consists of three thicknesses of 
1^-inch boards. The first thickness is laid diagonally across 
the opening. The ends resting on the rebates of the wall-plates, 



* Mahan's Civ. Bng., Wood's Ed., pp. 135 and 136 ; Barlow on the Strength 
of Materials, p. 183. 
f Tredgold's Principles of Carpentry, 1870, p. 91. 



BEAMS OF UNIFORM RESISTANCE. 203 

and rising about 2-J- inches higher in the middle of the room 
than at the sides. The second thickness is also laid diagonally, 
but square across the first, and the two well nailed together. 
The third thickness is laid parallel to the sides of the room, 
and the whole well nailed together. All the boards are grooved 
and tongued together, forming a floor 4J- inches thick. The 
strength of plates vary as the square of their thickness, and 
are equally strong to support a weight in the middle, whatever 
the extent of the bearing may be ; but when the load is uni- 
formly distributed, the strength varies inversely as the area of 
the space it covers.* 

* Emerson's Mechanics, sec. viii., prop. 73, cor. 5. 



204 THE RESISTANCE OF MATERIALS. 



CHAPTER YIII. 

TORSION. 

170. torsive strains are very commcn in machinery. 
In all cases where a force is applied at one point of a shaft to 
turn (or twist) it, and there is a resisting force at some other 
point, the shaft is subjected to a torsive strain. The wheel and 
axle is a familiar case in which the axle is subjected to this 
strain. To produce torsion without bending, a couple, whose 
axis coincides with the axis of the piece, must be applied to the 
piece. If only a single force, P, is applied, as in Fig. 108, the 
piece is pushed sidewise at the same time that it is twisted ; 
but the amount of twisting is the same as if the force, P, were 
divided into two, each equal \P, and each of these acted on 
opposite sides of the axis and in opposite directions, and at a 
distance from the axis equal AB, Fig. 108. For, the moment 
of the couple thus formed, is \P x 2 AB = P.AB, which is the 
moment of P. 



171. the angle of torsion is the angle through which 
a fibre whose length is unity, and which is situated at a unit's 
distoMce from the axis, is turned by the twisting force. It 
depends for its value, in any case, upon the elastic resistance to 
torsion, as well as upon the dimensions of the piece and the 
twisting force. The analysis by which its value is determined 
is founded upon the following hypotheses, which are approxi- 
mately correct. 

1st. The resistance of any fibre to torsion varies directly as 
its distance from the axis of the piece. 

2d. The angular amount of torsion of any fibre between any 
two sections, or the total angle of torsion, varies directly as the 
distance between them. 



TORSION. 



205 



3d. The resistance to torsion varies directly with the elasticity 
of the material. 

It is found by experiment, that these hypotheses are suffi- 
ciently exact for cylinders and regular polygonal prisms of 
many sides. They assume that transverse sections which were 
plane before twisting, remain so while the piece is twisted, but 
in reality the fibres which were parallel to the axis before being 
twisted are changed to helices, and this operation produces a 
longitudinal strain upon the fibres ; and this, in turn, changes 
the transverse sections into warped surfaces.* 

To find the angle of torsion : — 

Let I — AD = the length of the piece, Fig. 108. 
P = the twisting force ; 
a = AB = the lever arm of P ; 

a = aAb — the total angle of torsion, or angle through 
which Aa has been twisted ; 

= j = " The Angle of Torsion," — supposed to be small ,* 

f{Q> <f) = the equation of a transverse section, and 

G = the coefficient of the elastic resistance to torsion, which 




Fig. 108. 



is the force necessary to turn one end of a unit of area and 
unit of length of fibres through an angle unity, the vertex of 
the angle being on the axis of torsion, one end of the fibres 
being fixed and the twisting force being applied directly to the 



* Resume des Lemons, Navier, Paris, 1864, p. 276, and several other pages 
following. 



206 THE KESISTANCE OF MATEEIALS 

other end, and acting in the direction of a tangent to the arc of 
the path described by the free end. 

As a unit of fibres cannot be placed so that all of them will 
be at a unit's distance from the axis, we must suppose that the 
resistance of a very thin annulus, which is at a unit's distance, 
is proportional to that of a unit of section. The area of an 
element is 

pdpd(f>. 

The resistance of an element which is at a unit's distance 
from the axis is G multiplied by its area ; which expressed 
analytically is 

Gpdpd<p, 

and according to the first law 

Gp 2 dpd(f> = the resistance of any fibre 
whose length is unity, to being twisted through an angle unity ; 
and the moment of resistance = Gp s dpd<f> for an angle unity ; 
and for any angle the moment is, according to the second law, 

G0p s dpd<l> 

and the total moment equals the moment of the applied force, 
or moments of the applied forces ; hence 



L P5 



Pa = GQffp*dpd$ =GjI P 

where I p is the polar moment of inertia of the section. 

2tt 
For circular sections I p = I I p*dpd<f> = \in^ (199) 



Jo Jo 



G = M*™ (200) 



TORSION. 207 

1 72. the value of the coefficient g may be found 
from Equation (200). M. Cauchy found analytically on the 
condition that the elasticity of the material was the same in all 
directions, that G = f JK* M. Dulean found experimentally 
that G is less than f £J, and nearly equal £ J5J f and M. Wert- 
heim found 6r = f J^nearly.J M. Duleau's experiments gave 
the following mean values for G : % 

Value of Q. 
Pounds 

Soft iron 8,533,680 

Iron bars 9,480,917 

English steel 8,533,680 

Forged steel (very fine) 14,222,800 

Cast iron 2,845,600 

Copper 6,209,670 

Bronze 1,516,150 

Oak 568,912 

Pine 615,472 

Example. — If an iron shaft whose length is 5 feet, and diameter 2 inches, 
is twisted through an angle of 7 degrees by a force P = 5,000 lbs., acting on a 
lever, a = 6 inches, required G. The 7 degrees is first reduced to arc by mul- 

TC 7?T 

tiplying it by — , which gives a = — , and Eq. (200) gives, 

n 2x5000x6x60x180 _ __„ ... _, 
G = (8.1416)' x 7 ~ = 9 ' 697 '° 00 lbS ' 



173. TORSION PENDULUM — If a prism is suspended from its 
upper end, and supports an arm at its lower end, and two weights each equal 
•^•TTare fixed on the arm at equal distances from the prism, and the prism be 
twisted and then left free to move, the torsional force will cause an angular 
movement of the arm until the fibres are brought to their normal position, 
after which they will be carried forward into a new position by the inertia of 
the moving mass in the weights \W until the torsional resistance of the prism 
arrests their movements, after which they will reverse their movement, and 
an oscillation will result. 

Equation (200) readily gives : 

D irGr* 



* See Chapter IX. 

f Resistance des Materiaux, Morin, p. 461. 

% I? Engineer, 1858, p. 52. 



208 THE RESISTANCE OF MATERIALS. 

from which it appears that the torsional force P varies as the space (an) over 
which it moves. 

It is a principle of mechanics that the moving force varies directly as the 
product of the moving mass multiplied by the acceleration. Hence, if x = (an), 
the variable space, t = the variable time, M = the mass moved, and observing 
that t and x are inverse functions of each other, and the above principle of 
mechanics gives the following equation (neglecting the mass of the prism) : — 

T, r d 2 x _ KGr* 

Multiplying both members by the dx, gives 

W dxd 2 x _ 7r#r 4 
g db* ~ ~2W XdX ' 

where W is the weight of the mass moved, and g is the acceleration due to 
gravity. The oscillations commence at the extremity of an arc whose length 
is *, at which point the velocity is zero. The integral of the last equation 
between the limits s and x is 

dx*_ 7cGgr* ' 

dt*-%Wla* {S X) ' 

A second integral gives 

t= \/-VGg^ L Sm TJ = V^^ r 
which is the time of half an oscillation. For a whole oscillation : 



"W&* 



2t- 

Gff 

This is essentially the theory of Coulomb's torsion pendulum. A torsion 
pendulum was used by Cavendish in 1778 to determine the density of the 
earth. (See Royal Philosophical Transactions: London, Vol. 18, p. 388.) He 
found the mean density of the earth by this method to be 5.48 times that of 
water, or according to Hutton's revision, 5.42. 

Reich, by aid of a mirror apparatus, afterwards found it to be 5.43. Bailey 
found by experiments on a larger scale 5.675. Reich repeated his experiments 
and found 5.583. Other methods gave a value somewhat larger than these, 
but the mean result shows that the mean density of the earth is about 5| times 
that of water. — See Bailey's Experiments, London, 1843. 



1 74. rupture by torsion. — The resistance which a bar 
offers to a twisting force is a torsional shearing resistance, and 
in regard to rupture, the equation of equilibrium is founded 
upon the following principles : — 

1st. The strain upon any fibre varies directly as its distance 
from the axis of torsion ; and 



T0KSI0N. 209 

2d. The sum of the moments of resistance of the fibres equals 
the sum of the moments of the twisting forces. 

Let J= the modulus of torsion, that is, the ultimate resist- 
ance to torsion of a unit of the transverse section which is most 
remote from the axis of torsion. It is the ultimate shearing 
resistance to torsion, but may be used for any shearing strain 
which is less than the ultimate, 

d x = the distance of the most remote fibre from the axis of 
torsion, 
f (p, <f>) — the equation of the section, 
P = the twisting force, and 
a = the lever arm of P. 

I<p = the polar moment of inertia of a section. 
Then pdpd<j> = dA = the area of an element of the section ; 
Jpdpdfy = the shearing strain of the most remote 
element ; and, by the first principle given 
T above, 

— pdpd(f> = the shearing strain of any element, which 

^ is at a unit's distance from the axis of 

torsion, and from the same principles we 

have 

—p 2 dpdcj> = the shearing strain of any element, and 

1 this, multiplied by the distance of the 

element, p, from the axis, gives 

-j p z dpd<f> = the moment of resistance to torsion. 
ELence, according to the second principle we have 

pa = iff^ - if^-i 1 * w 

For circular sections, we have already found, Eq. (199), 

For square sections, whose sides are h, we may find * 

Ip = % ¥, and d 1 — h \/£ 

__ , 1 

* We hsLveJp 2 dA =j(® 2 +y 2 )dA =Jx 2 dA +Jy 2 dA, that is, the polar mo- 
ment equals the sum of the rectangular moments, the origin being the same 
14 



210 THE RESISTANCE OF MATERIALS. 

175. practical formulas. — Equations (199) and (201) 
give for cylindrical pieces, observing that d t = r, 

Pa = i7rJr* .'.J= ^? (202) 

If cylindrical pieces are twisted off by forces which form a 
couple, and P, a, and r measured, the value of e/"may be found 
from Equation (202). Cauchy found J = £ i?,* which is con- 
sidered sufficiently exact when a proper coefficient of safety is 
used. Calling J — 25,000 pounds for iron, and using about a 
five-fold security; and J= 8,000 pounds for wood, and using 
about a ten-fold security, and we may use for 

Round iron shafts (wrought 

or cast), diameter = -^ fyPa 

Square iron shafts (wrought ( (203^ 

or cast), side of the square = -fa ItyPa! 
Square wooden shafts, 

side of the square = \ \/Pa 

The dimensions given by these formulas are unnecessarily 
large for a steady strain, but shafts are frequently subjected to 
sudden strains, amounting sometimes to a shock, and in these 
cases the results are none too large. 

Practical formulas may also be established on the condition 
that the total angle of torsion shall not exceed a certain amount. 
Making G — f E, and solving (200) in reference to r, and we 
have for cylindrical shafts, 



V 



16 Pal 



SirEa 

and similarly for square shafts, 



=v 



16 Pal 



E* 



in both cases. In this case the origin being- at the centre of the square, we 
have fx'dA=fy' 2 dA .\lp =^J % y r dA = 2x ^b A (see Eq. (51) ). 
* Resume ,/es Lemons, Navier. Paris, 1856, pp. 193-203, and p. 507. 



TORSION. 211 

In these expressions P should not be so great as to impair 
the elasticity, — say for a steady strain P should not exceed the 
values given by Equation (203). 

If a° is given in degrees, it is reduced to arc by multiplying 

IT 77* 

it by — — so that a = ^r«° ; hence the preceding equations be- 
come : for cylindrical iron shafts, 

r = 3.14//^ (204) 

and for square iron shafts, 

d = 5.51 x 4 /— (205). 



Examples. — 1. A round iron shaft 15 feet long, is acted upon by a weight 
P = 2,000 lbs. applied at the circumference of a wheel which is on the shaft, 
the diameter of the wheel being 2 feet ; what must be the diameter of the 
shaft so that the total angle of torsion shall be 2 degrees ? 

If the shaft is cast-iron E = 16,000,000, and 



o * flOO 4/2000xl2x 



15 x 12 . fln . , 

ooo- = 3 - 69mclies ' 



2. A round wooden shaft, whose length is 8 feet, is attached to a wheel 
whose diameter is 8 feet. A force of 200 lbs. is applied at the circumference 
of the wheel, what must be the diameter of the shaft so that the total angle 
of torsion shall not exceed 2 degrees ? 



■ = d— 6.284/ ~ 



2 00 x 4 x 12 x 8 x IS 
2x2,000,000 



1 75a. RESULTS OF WERTHEDI'S EXPERIMENTS A f eW 

years since M. G. Wertheim presented to the French Academie 
des Sciences an exhaustive paper upon the subject of torsion, 
the substance of which was published in the Annates de Chimie 
et de Physique, Yol. XXIIL, 1st Series, and Vols. XL. and L., 
3d Series. These articles would make a volume by themselves, 
and hence we will content ourselves at this time with present- 
ing his 



212 



THE RESISTANCE OF MATERIALS. 



CONCLUSIONS. 



When a body of three dimensions is subject to torsion the 
following facts are observed : — 

1st. The torsion angle will consist of two parts, one tempo- 
rary, the other permanent; the latter augments continually, 
though not regularly. 

2d. The temporary displacements augment more and more 
rapidly than the moments of the applied couples, and the in- 
crease of the mean angle, which in hard bodies continues until 
rupture, in soft bodies continues only to the point where the 
body commences to suffer rapid and continuous deformation. 

3d. The temporary angles are not rigorously proportional to 
the length, and, all else being equal, the disproportionality in- 
creases in measure as the bar becomes shorter. 

4th. In all homogeneous bodies, torsion caused a diminution 
of the volume, which is proportional to the length and square 
of the angle of torsion, and each point of the body, instead of 
describing an arc of a circle, follows the arc of a spiral. The 
condensation of the body increases from the centre to the cir- 
cumference. 

5th. In bodies with three axes of elasticity, the change of 
volume and resistance to torsion are functions of the free axes, 
and the relation between them may be such that the volume 
will augment. 

6th. Circular or turning vibrations of great amplitude are 
difficult to produce, and as small angles of torsion only are 
used, the preceding conclusions apply to this case. 

7th. Rupture produced by torsion usually takes place at the 
middle of the length of the prism ; it commences at the dan- 
gerous points, and operates by slipping in hard bodies and by 
elongating in soft ones. 

8th. With regard to the influence of the figure and absolute 
dimensions of the transverse sections of the bodies, we derive 
the following conclusions : — 

9th. In homogeneous circular cylinders the diminution of the 
volume is equal to the original volume multiplied by the prod- 
uct of the square of the radius, and the angle of torsion for a 
unit of length (the angle being always very small). Further, 



TORSION. 213 

uncbr torsion the radius of the cylinder equals the primitive 
radius multiplied by the sine of the angle of inclination of the 
helicoidal fibres. This last gives a means of calculating the 
diminution of volume. But in reality the twisted cylinder 
takes the form of two frustra of cones joined at the smaller 
bases ; and although this does not sensibly affect the theoretical 
results for long cylinders, yet it deprives our formulas of all 
their value in ordinary practical cases. 



214: THE RESISTANCE OF MATERIALS. 



CHAPTEK IX. 

DISTORTIONS. 

176. any change: of form of a solid due to 
foreign forces is a distortion. — Several changes have 
been considered, separately and singly, in the preceding chapters, 
such as extension, compression, bending, torsion, and transverse 
shearing, but we will find that in all cases one of these distor- 
tions is accompanied by some other one. In all elastic bodies 
the particles move more or less freely under the action of the 
straining forces. 

The phenomenon of elasticity is nothing more than the 
action of the attractive and repulsive forces of the molecules 
of a body upon each other. When a force is applied to a body, 
its effect is transmitted from particle to particle by the internal 
forces, until it meets and is held in equilibrium by a force 
applied at some other part. 

The Mathematical Theory of Elasticity is considered in three 
parts, the relation of stresses, the relation of strains, and the 
relations of stresses to strains.* We shall here consider only 
such principles as pertain immediately to the problems under 
consideration. 

177, measure of slipping.- -If the section bdbe forced 
into the position qfby the slipping (transverse shearing) of bd 
upon ac, the amount of the movement per unit of length will 
be measured by the angle baq, which for small displacements 
will be measured by the tangent of the angle. 

Let g be the tangent of baq = -| = bq when ab is unity. 

* M. Lame's Lemons sur la TMorie Matliematique de V Elasticity des Corps 
Bolides^ Paris, 1852. Resistance des Corps Solides, par Navier. Troisieme edi- 
tion, 1864. 



DISTORTIONS. 



215 



The resistance to this shearing will evidently vary as g, and 
also as the elastic resistance of the material, and if the resist- 
ance be evenly distributed over the transverse section it will 
also vary as this section. 




Fig. 109. 

Let P = the tangential force, that is, the force which acts in 
the plane bd, 
JE S = the coefficient of transverse elasticity, 
A = the area of the transverse section ; then 

P = E 9 Ag (206) 

If A = 1, we have j> = E 8 g, which is the intensity of the 
stress. 

When flexure is involved, we shall find that the shearing 
stress is not evenly distributed over the section. It is evenly 
distributed when bd is consecutive to ac, or when the area is 
small it may be considered uniform. 

" Letting fall the perpendicular ch from c upon aq, and we 
have 

ah _ hq 
ch ab 

hence, the transverse slipping in an amorphous body is accom- 
panied by an equal longitudinal one, for we consider that the 
effect is the same as if ab had slipped over cd, an amount equal 
to ah. 

Produce the diagonal ad and describe an arc fe which shall 
pass through f, having the centre at a, then will de be the elon- 
gation (or dilation) of the diagonal ad ; and in a similar way we 
may find the contraction of the diagonal cb. 



9\ 



216 THE RESISTANCE OF MATERIALS. 

If i = the dilation per unit of length, we have 

• __ ^ e 
~~ ad 

By similarity of the triangles def and acd we have 

<^_-,_P ecu df ca 

i — — ~ a ^ ad' cd ad 

which is a maximum when cd = ac, for which case we have 

i = %9 (207) 

or the maximum dilation is one-half the slipping. Similarly, 
the maximum -contraction of eft takes place when it is the diag- 
onal of a square. 

We see that if we have two equal stresses in opposite senses, 
one a pull along ad, and the other a push along eft, whose di- 
rections make a right angle between them, the resulting distor- 
tion is equivalent to one-half of a simple shear of the same 
intensity on a plane at 45 degrees with either of the others. 

Limit of the Slipping. 

If R' = the elastic limit of the strain ; 
E = the coefficient of elasticity ; and 
i' = the elongation produced by It', we have 

., R' - 2R' 



178. RELATION BETWEEN LONGITUDINAL AND, LAT- 
ERAL stresses. — When a body is subjected to a pull there is 
a lateral contraction, as shown in Fig. 116. The relation between 
these stresses, for bodies which are not homogeneous, is com- 
plex, but it is one of the questions which is considered in the 
Mathematical Theory of Elasticity. But for a solid whose 
elasticity is the same in all directions — called an isotropic body 
—the relations are comparatively simple. First consider the 
case in which the straining force acts in any direction. 




DISTORTIONS. 217 

Let jp = a stress acting in any direction due to any cause, 
jPxz* n= the component of this pressure upon a unit of sec 
tion resolved normally to a plane which is perpen- 
dicular to x ; 
^>yy = the normal component on a 
unit of section which is nor- 
mal to y ; 
Pzz = similar component in regard 
to z\ 
A = the coefficient of direct or lon- 
gitudinal elasticity, which 
expresses the relation be- ■ FlG - 1 ^^* 

tween the longitudinal strains and the normal 
stresses ; 
B = the coefficient of lateral elasticity, which expresses 
the relation between the longitudinal strain (a 
push or pull) and the stresses at right angles to 
the strains. It expresses the resistance to lateral 
contraction, 
&„ S v , S g = the elongations in the directions of the axes x, y 

and s for a unit of length. 
We then have, when the body is perfectly amorphous, or 
isotropic,t 

* This notation was first used by Cauchy and Corioles in discussions upon 
the Theory of Elasticity. The first sub-letter indicates the normal to the 
plane, and the second one the direction of action in that plane. Thus^xy 
indicates a pressure upon a unit of area which is perpendicular to #, and in a 
direction parallel to y. 

\ Let Aa be an elementary section, 

r = Mm = the distance of any molecule, m to the right of Jf, 
R = the force exerted by one molecule upon another at the distance and 

in the direction of r, 
n the number of molecules contained in a unit of the body in the vicinity 
of if. 
Take the origin of coordinates at M, x being taken perpendicular to the sec- 
tion Ja, z vertical, and y perpendicular to x z. 

The total action of all the molecules which are distant Mm = r of the mole- 
cules on the right of M upon those at the left, is the same as if the whole mass 
of the cylinder at the left, whose length is r, were concentrated in the section 
at M acting upon the lamina at m ; which is the same as if it were concentrated 
in the point M, and the lamina in the molecule m. We have 



218 THE BESISTAHCE OF MATEEIA1S. 

jp„ = A8 V + £(S Z + S X ) (208) 

p. = AS. + £(B X + S p ) 
Suppose that the strain is parallel to x, then jPyy=0 and ^=0. 

Aa . x = the volume of the cylinder, 
n Aa . x = the number of molecules, 
Bn Aa .x — the sum of all the actions parallel to r % between Mm, 

Bn Aa .x- = the resultant normal to the section. 
r 

If we consider r as variable, and for each new value of r we substitute a 

proper corresponding value of R (which might be called "R u R 2 , R 3 , etc.), we 

shall have a series of corresponding expressions all of which will have the 

same form as that given above, hence we have for all the forces which cross 

the section Aa 

— x 2 
AaSB- 
r 

x 2 
in which S applies to all values oinB- from zero to r, when the expression 

does not reduce to an insensible quantity on account of the rapid decrease of 
its value as r increases. The relation between R and r is not known, but we 
may assume that the resistance offered by the elastic forces above those which 
in the natural state are in equilibrium, when disturbed by an extraneous force, 
is proportional to the small increase of distance, dr, as we found in Chapter I. 
Let Bi be the derivative of B in respect to r, then will the stress on a unit be 

8B x drx, and 
x 2 

will be the resolved component of the stress. 
Let S x , S y and 5 3 be as given in the text, then 

6 X - x — the projection on r of the x component of the elongation, and 

T 

similarly for y and z ; 
hence, neglecting all differences above the first, we have 



hence 
Similarly 



dr = n x - +hz +8 3 - 
r r t 



Pxx = tixSBx ^j -f- dy SBt. -p- + 5 S SBi ■—$- 
«/ 4 y 2 z 2 v 2 x z 



DISTORTIONS. 219 

Also the lateral compressions will be the same in the directions 
of y and s, and hence B y = S z . For this particular case let 
&x = h.) $ v = S z = i 2i and we have 

p^ = Ai t + 2^ 2 (208a) 

= Ai 2 + Bi 2 + Bit (209) 

By elimination we find 



A + B 

AB- 

-B 



h ~A*+AB-2B*^ 



\P* 



' A 2 + AB- 2B 2 

Hence i 2 is negative compared with %, as it should be, since 
a longitudinal pull produces a lateral compression. 

The value of % when p^x is unity is called the coefficient of 
direct pliability, and i 2 the coefficient of lateral pliability. 

Returning to the equations, and we find 

a _ H^-h 



i} + %4 — 2^ 2 



fr 



Or, since i % is negative, we have for the numerical values of 
A and B when p^ is unity, and i x and i 2 are both used as posi- 
tive numbers ; 

A — ^ ~ ^ 

ii — hh — 2^2 



s 4 2 2 # 2 s 2 '?/ 2 

But on account of the isotropic character of the solid, the expressions 
which are similar will have the same value, hence 

SB, £ = SB, £ = SB, £ = 4 (say), 

^Si^x — 5- = iSi?! — =— = SMi — 5- = S ; 

which reduces the preceding equations to those in the text. 

It is more common in the investigations in molecular mechanics to prove 
at once a relation between A and B. The preceding is a special solution. 



THE RESISTANCE OF MATERIALS. 



A — iih — %i\ 

To find the relation between A and B requires a series of 
experiments, or a further consideration of molecular actions. 
Since the solid is isotropic, we shall here assume that the re- 
sistance to a shearing stress is the same in all directions, and 
since the lateral movement only takes place by shearing, we 
will here assume that the coefficient of lateral elasticity is the 
same as that for transverse shearing, or torsional shearing, or 
longitudinal shearing. 

Generally let C = the coefficient of transverse elasticity ; 
the particular values of which will be for an isotrope, 

O = B = E t = ^ (see Eq. 206) = G (see Torsion). 

We have (as shown below), 

A - B = 20 = -— t-v p^ = 2B 

H — h 

i 

Pxx numerically ; 



h. + V 
/. C= i{A - B) (210) 

To prove this, take the case of two equal stresses acting at 
right angles with each other, in which one is a pull, and the 
other a push ; the former being parallel to xx, and the latter 
parallel to yy. Since the body is isotropic, the contraction in 
the direction of z produced by j?^ will equal the expansion 
caused byj?yy, and hence $ g , Eq. (208) will be zero, and^> zz = 0; 
hence the third of Eq. (208) gives 8 X = - $„, and Eqs. (208) 
become 

<p xx = (A-B)S x 

p Yy = {-A + B)B x =-p^ 

The intensity of the transverse shearing is, Eq. (206), j> = 
E s g = Cg which for this case is, Eq. (207) 

J> =i>xx(or - Py7 ) = 0.2i = 208 x 

hence A- B = 20. 



DISTORTIONS. 221 

Assuming, as above stated, that the lateral elasticity for art 
isotrope is the same as the transverse, and we have 

A-B = 20=2B 

.\ A = SB 

which in the second of Eqs. (209) gives 

h=-¥u 
that is, the lateral contraction of an isotropic solid "under the 
action of a direct stress is J as much per unit as the longitu- 
dinal extension for the same unit. 
If h = the breadth, 
d = the depth, 
I = the length of a prism, 
V= the volume before elongation, 
Fi= " after " , and 

P = the pulling force. 

Then the elongation is 

P7 
\=-^(Eq.(l)) 

and i x =~j- 

We also have 

V = Ml 
.-. V % = b (1 - 4) d (1 - %) I (1 + h) 

= Ml (1 + ii — i 2 ~ h) nearly. 

If i 2 = i 6 exceeds %% the expression becomes negative, or 
there would be a diminution of the volume, which is absurd ; 
hence this may be considered a superior limit of the values of 
i 2 and %. If i 2 = i% = i%, we have 

V x = Ml (1 + ¥i) = !>d(l + i\) 

This solution shows, that on the convex side of a bent beam 
there will be a lateral contraction, and on the concave side 
there will be a lateral expansion. The elongation per unit of 
length for a rectangular section whose 1 readth is b, and depth 
d, is, Eq. (45), 

2p 



222 THE RESISTANCE OF MATERIALS. 

and the lateral contraction per unit will be 

and the expansion on the opposite side will be the same amount 
The total contraction will be 

M 

8p 

By this means we find that the lateral sides prolonged afteT 
flexure will meet at a point 

from the neutral axis opposite to the convex side. The sections 
which were rectangular before flexure become trapezoids after 
flexure. 

Substitute the values of A and i 2 in Eq. (208#), and we find 

5 5 

But ±^- = JT, the coefficient of longitudinal elasticity, 

;.G=\e. (211) 

or the coefficient of elasticity for slipping (whether transverse, 
lateral, longitudinal, or torsional), in an isotropic solid is f of 
the coefficient of longitudinal elasticity. 

But G= B = iA; 

that is, the coefficient of longitudinal elasticity is -| of that for 
the direct normal stress. 

It will be seen from the equations, that the quantity E is a 
result of all the distortions, being the elongation which results 
from the yielding both longitudinally and laterally. 

Eeturning again to Eqs. (208) and we have 

ifea +i?yy +iO = i(^ + 2B) (B x + 8 p + 8.) 

The coefficient -J- ( A + 2B) expresses the relation between the 



DISTORTIONS. 223 

mean direct stress and the cubic strains, and is called the cubic 
elasticity, or elasticity of volume. 

The coefficient of transverse pliability is _ = = 2 (% -f- ^) 

C 

(numerically). 

The coefficient of cubic compressibility is 

D = A 2£ = 3 (h. - 24) (numerically). 

The following are the coefficients for Crystal, as deduced 
from the experiments of M. Wertheim. — (Annales de Chemie, 
3d Series, vol. xxiii.) 

A 8,522,600 
B 4,204,400 
G 2,159,100 

i 5,643,800 

I 5,747,000 

^ 0.0000001740 
% 0.0000000575 
c 0.0000004631 
D 0.0000001772 



179. SHEARING STRAINS IN A BEAM WHICH IS BENT 
BY TRANSVERSE STRESSES. 

In the discussions of the problems of flexure, the longitudi- 
nal elements were treated as if they produced no action upon 
each other, and were simply subjected to the laws of extension 
and compression. 

If we conceive that the beam is composed of thin horizontal 
laminae, and each had an uniform strain from one end to the 
other, there would be no slipping between the adjacent ele- 
ments. This case is realized when the neutral axis is the arc 
of a circle, as when a beam is supported at its ends and is 
loaded with equal weights placed at equal distances from the 
supports. But in all other cases there is necessarily a slipping. 



224 



THE RESISTANCE OF MATERIALS. 



In Fig. Ill, the element DC is not subjected to any tensile 
strain at O, but from that point the strain increases to Z?, where 
it will be a maximum. The strain can be unequal only by 




Fig. 111. 

some adjacent element or elements taking off (so to speak) some 
of the pulling force, and the element must slide upon the 
adjacent one. 

The moment of P in reference to e is Px, and in reference 
to/* it is P{x 4- dx), and the difference between these, or Pdx, 
is the moment of the shearing force atyin reference to e, and 
hence the shearing force is P, as given in Article 93. It re- 
mains to determine the law of distribution of this stress. 

The strain on a unit of section qn is, according to Eq. (46), 

E 

p 
w 

in which substitute the value of — from Eq. (49), and we have 

P 



p = ?LXPx. 



(212) 



and for a width mm! (Fig. 112) = U the strain is 
and for a depth qn = dy it is 



1L USPx 



$ Udy = -^y Tidy. 



DISTORTIONS. 



225 



The strain at jpra is found in the same way, and the difference 
between the two is the excess of strain at m over that at n. 




Fig. 112. 



It is evidently the differential of the preceding expression in 
which p and ZPx are the only variables (the beam being pris- 
matic). 



.*. dp Tidy 



d%Px ^ , 2 P. dx ^ j 



i 9 * i 

(See Article 93 in regard to the last reduction.) 

This is the value of the longitudinal shearing along rrm = dx 
due to the element mq. But the shearing due to all the ele- 
ments between c and n will be 

SPdr rdl 



1 J y 



(ch being the distance of the extreme fibre from the axis). This 
is the expression for the force which tends to move the volume 
acnm along the line mn. This divided by the area mn m'm = 
Udx gives the shearing strain on a unit of section, or its inten- 
sity, which is (say Gg, as stated on the next page.) 



Gg 



^wjy^y 



.(213) 



which at the centre becomes 



Cfa = 



■zp r<k 



ib 



"0 



Uydy. 



.(214) 



m which h is the breadth on the neutral axis. 
15 



226 THE RESISTANCE OF MATERIALS. 

But it was shown in Article 177 that a longitudinal shearing 
is accompanied by an equal transverse shearing. Hence, Eq. 
(213) gives the intensity of the transverse shearing at a point 
whose ordinate is y. 

If XP becomes zero, as it does in the case of a Couple, Eq. 
(213 s reduces to zero, or there is no transverse shearing. 

The par.: / Uydy is the statical moment of aalmm' in 
J y 

reference to the neutral axis. 

We see from this that the transverse and longitudinal shear 
ing is zero at the upper and lower sides, and increases towards 
the neutral axis, at which place it is a maximum. We may 
conceive of this condition by supposing that the beam is built 
up of successive layers, each succeeding one being free, but 
adding to the shearing of all the preceding ones between it and 
the neutral axis. Equation (213) is not exact except for rectan- 
gular cross sections, for when the cross section is elliptical or 
otherwise curved, the free surface is at variable distances from 
the neutral axis. 

If rrik is normal to mp, and 
?nr is parallel to ot, then 

g = tangent of the angle fcmr, the amount of slipping, 
G = coefficient of shearing elasticity, = E 8 for isotropes, and 
Qg = the resistance to shearing per unit, as used above. 

The mean intensity is 

SP 
A 

Hence, the ratio of the maximum shearing is to the mean as 
Gg A A f dl n 7 

which depends entirely upon the form of section. 

If the cross sections are rectangular, Equation (214) gives 

'id 
ZP I , , SP 



I, 



G v°=&m< A w = tu (215) 



DISTORTIONS. 227 

which is the maximum intensity of the shearing, and is | of 
the mean. The total shearing is 

If the beam be fixed at one end and loaded at the free end, 
SP = P, and hence the total force which at all points along 
the neutral axis tends to push the upper half of the beam along 
the lower half, and which should be resisted by the cohesion 
of the elements, will be per unit of length 

<&» = *£. 

For any portion aa'mm! of the rectangle we have 

G * = Bjy ^=^[>-^]- (216 > 

OI G(g - & ) = 6 -fff (216a) 

that is, in rectangular beams the difference between the shearing 
at the neutral axis and any other point above or below it, 
varies as the square of the ordinate. 

The total transverse shearing in the cross section is 

which by reduction gives %P, and agrees with Article 93. 

It now appears evident that when there is longitudinal 
shearing, the transverse sections which were originally plane, 
will not remain plane during flexure. 

To find the Equation of the Curve aob for rectangular 
beams, 

Erect oe normally to ot at o, and let 
y = the ordinate os, and 
x = the abscissa ms, 
mr is parallel to ot, 
mk is normal to mp, and 
g = tangent kmr = the slipping. 



228 THE RESISTANCE OF MATERIALS. 

-j- = the inclination of rrvp to oe, = g plus a small inclina- 
y tion which mn may have in regard to ot, which is 
produced by the lateral contraction due to the 
longitudinal extension. 
The elongation per unit of length is 

\ (Eq.(45)) 
P 

and the contraction for an isotropic solid is J — , and hence the 

P 
contraction between o and m is 



i ; 



V Xlrl -Vl -X 2 ^ PX 

± P d y- 8p~ 8 EI 



The distance rno will be less than tn by an amount which 
will not differ sensibly from the differential of the preceding 
expression. Differentiating and dividing the result by dx 
gives for the tangent of the inclination 

8EI 
Hence, for rectangular beams we have 



dx_ SP 2 _6ZP 

dy 9 ^ SEI V ~ Gbd" 



\ld*-f + ^i\ 



Integrating, observing that x = for y = 0, and we find 
32P 



0== 4OT 



\_d \ IE) 3^ 3 J 



Each half of the curve aob is therefore a parabola of the 
third degree, and of the same order as the curve AB. 
For the abscissa ae, at the upper surface, make y — \d. 

•~ = Ksi 8+ 3 = *§— * (217) 

We here have the peculiar result that the total effect of the 
longitudinal shearing at the surface is independent of the depth 
and length. 

If G = %E, we have 



DISTORTIONS. 229 

a, = l l&[ 5 ^~ 6y8 ] (217a) 

if SP = 0, as in the case of a Couple, x will be zero for all 
values of y, and hence the section will be plane. 

The curve is normal to the neutral axis, or parallel to oe, 

doS 

where -=- = 0, or at the point whose ordinate is 
dy 

y = 0.52d, 

or at a short distance above the upper surface. Were there no 
lateral contraction it would cut the upper surface normally. 

The algebraic curve, if continued, will cut the axis of y at 
two points, one at y = 0, and the other at \/^ d, or at 0.91 d 
nearly. 

Example.— If 2P=P= 4,000 lbs., G = 8,000,000 lbs,, 5 = 1 inch, d = 12 
inches, required the ordinate ae. 

Ans. ae = ^ - - inch, nearly. 

It will be seen from the preceding equation, that all the 
transverse sections in prismatic beams which were originally 
parallel, will be of the same form when the beam is loaded at one 
point only ; and that it will be modified at the different points 
of the beam if the value of the shearing stress, 2P = S s varies. 

180. INCREASED DEFLECTION DUE TO TRANSVERSE 
SHEARING (OR SLIPPING). 

The slope of any element due to the transverse shearing, 
Eq. (213), is 

XP M • 

tongfanrz=g=Qjpl t TJydy, 
*y y 

which must be added to the slope due to bending by flexure. 
The increased deflection for a length I will be 

h'=.lg 

if $ is constant, but if it is variable, 



i'= f X gdx; 



h' = J gdx / or by substituting the preceding value of g, we 
have 



THE RESISTANCE OF MATERIALS. 

GiuJ y *** oiuj >wr. ■ ( 217J ) 

If the beam is rectangular, the limits are and \d ; 

" d -• Gbd> 

hence the deflection varies directly as the moment of the 
bending forces and inversely as the area of the transverse 
section. 

If the beam is fixed at one end and loaded at the free end, 

Hence the total deflection will be (Eq. (57) ) 
Id 



SEP + 2G J 



If it be uniformly loaded 

If it be supported at its ends and loaded at the middle, 
XPx — iPl, hence 

6 -^Gbd' 

and hence the total deflection will be (Eq. (73) ) 

PP P7 

+ #~ (218) 



Prof. W. A. Norton, of New Haven, Ct., detected the exist- 
ence of the last term, involving the deflection due to transverse 
shearing* directly from experiment. He assumed that this 
stress is uniformly distributed over the transverse sections, and 
deduced an equation of the same form as the preceding, the 
only difference being in the value of the coefficient, G. Al- 
though the stress increases from the outer surfaces to the neu- 
tral axis, as we have seen in Equation (216), yet the resistance to 

* Van NostrancPs Eclectic Engineering Magazine, Vol. 3, p. 70, 1871. 



DISTORTIONS. 231 

deflection of rectangular beams varies according to the same 
law, as if it were evenly distributed. 

Prof. Norton's experiments were made upon prismatic white 
pine sticks, and the mean of a large number of experiments 
gave 

^=1,427,965 108. 

and A aa 0.0000094: lbs.* 
86- 

.-. G — 40,000 lbs., very nearly. 

This is only -£j of the value of E. So small a value at first 
caused a doubt as to the applicability of the formula to fibrous 
beams. But the experiments of Chevandier and Wertheim, 
p. 17, give for white pine the coefficient of elasticity in the 
direction of the radius 97.7 kil. per sq. millimetres, or 135,950 
lbs., and in the direction of the tangent to the layers, 40,680 
lbs., from which we see that the value given above may be 
correct for the material used, and for the position in which it 
was used. 

Prof. Norton also informed the author that there were dis- 
crepancies in the experiments which he was not able to account 
for at the time , but that, in the light of Chevandier and Wert- 
heim's experiments, he was of the opinion that they were 
mainly clue to the position of the layers in the specimens, as 
some might have been horizontal, others vertical, and still others 
inclined, when the experiments were made. Duleau's experi- 
ments, Article 173, gave for the coefficient of elasticity for 
torsion (perpendicular to the direction of the fibres) 615,472 
lbs. For fibrous bodies there is no simple relation between 
the coefficients of elasticity in the different directions. For 
cast iron, wrought iron and steel it is generally assumed that 
the coefficient of shearing elasticity is \ that for longitudinal 
elasticity. 

Equation (218) may be written 



I ~ ri ~Gi*\ ( } 



'±EW 



^-pr is represented by 'C in his equation. 
o6r 



232 THE RESISTANCE OF MATERIALS. 

from which we see that the shorter the beam compared with 
its depth, the greater is the deflection due to transverse elas 
ticity compared with that due to the direct elasticity. In very 
short beams nearly the whole deflection may be due to shear- 
ing, while in long ones it may generally be neglected. 

181. DEFLECTION DUE TO IiONGITlHDINAIi SLIPPING.- 

If the lamina were free to slip upon each other, as we have be- 
fore illustrated by a pile of boards, they would retain their 
original length, and the deflection would be much greater than 
if there were no slipping. If the elements were held together 
by cohesion, but had no longitudinal shearing elasticity, but 
had, as now, a direct longitudinal elasticity and a transverse 
shearing elasticity, Equation (218) would give the deflection 
for rectangular beams supported at their ends and loaded at 
the middle. If the sections remained plane and were forced 
past each other, as in Fig. 26, without bending by flexure, 
then the total deflection would be given by Formula (217&). 
But there is a longitudinal shearing stress at every point where 
there is a transverse shearing, and the elasticity of the material 
permits a corresponding longitudinal shearing strain, and hence 
there is slipping, and the longitudinal elements are independent 
of each other to just the extent of the slipping and no more. 
When the longitudinal shearing elasticity is the same as the 
transverse, Equation (217) shows the total effect in a cross sec- 
tion of the shearing in a rectangular beam. 

Join o and <2, Fig. 112, with a straight line, and conceive that 
the beam turns about o so as to produce an opening ae at a. 
This would cause a deflection which we call d8, and by similar 
triangles we have 

ae : \d : : do : x 

,\ do = —=-x 
d 

=8GFdl 8 + -wJ x =md xne ^ 

Although this is not the correct expression due to longitudi- 
nal slipping, yet we may safely assume that it is proportional 



DISTORTIONS. 233 

to it. The longitudinal shearing elasticity may differ from the 
transverse. If then G x be the coefficient of longitudinal shear- 
ing elasticity, and r the ratio between the preceding expression 
and the true one, then the expression for the deflection due to 
this cause for an element distant x> will be 

rGJ>d X ' 

But the total deflection will be the sum of the expressions which 
result by giving to x all possible values from x = to x = the 
length considered; 

which for a rectangular beam fixed at one end and loaded at 
the free end, becomes 

2rGJ>d' 
and if supported at its ends and loaded at the middle, 2 J? = 
JP, and 

a- Pf 

16rGM 
This added to Equation (218) gives for the total deflection of a 
rectangular beam which is supported at its ends and loaded at 
the middle, 

A- Pl% FP + SFl 

" tJZbd* 16rG x bd 8Gbd 



' B = fG< 



[m + ik+A] (ai9a > 



4M1 Ed* 



NOTE. — As these pages are passing through the press, the author has re- 
ceived the following note from Professor Norton, which I am pleased to insert 
in this place, although I do not agree with his theoretical views. The fact, 
however, that his new formula represents the results of his experiments so 
accurately makes it worthy of serious consideration : 

' 1 1 find that the entire series of experiments which I have made on the de- 
flection of pine sticks, and iron and steel bars, loaded at the middle and rest- 
ing on supports, are represented with great accuracy by the following formula '. 

A = O* + I. PP 



bd ' "Affld* 
in which I denotes the length, b the breadth, and d the depth of the rectan* 



234 THE RESISTANCE OF MATERIALS. 

gular stick or bar, P the load at its middle, i^the coefficient of elasticity, and 

PL 6 
C a constant coefficient. The old formula, A — ■ , „ 7 ,.. involves two laws, 

^Ebd A 

which it appears from my experiments are very wide of the truth, unless the 

ratio of the length to the depth is large. These laws are, that the deflection is 

directly proportional to the cube of the length, and inversely proportional to 

the cube of the depth. The coefficient of elasticity, E, as determined with it, 

from any observed deflection, //, is ordinarily much too large. For example, 

it gives me for the wrought-iron bar I have used, as the value of E, 36,833,- 

000 lbs., and for the steel bar 37,066,000 lbs. 

" The formula which I before obtained is 

~ bd + 4Ebd» 

This, with certain values of E and (7, represented my experiments with white 
pine sticks much more satisfactorily than the old formula just referred to ; 
but the experiments on the deflection of wrought-iron and steel bars have 
shown it to be faulty — whether regarded as an empirical formula, or from 
the theoretical point of view — though it gives results that approximate more 
nearly to the truth than the one in general use. 

" In the new formula the first term represents the portion of the linear de- 
flection resulting directly from the shearing stress, propagated from the mid- 
dle of the bar to either point of support. This I had before conceived to be 
proportional to the length of the bar, but it appears, on careful consideration, 
that it is proportional to the square of the length. This may be seen by 
taking the case of a bar fastened at one end and loaded at the free end, and 
reflecting that when any one material section slips on the next one on the side 
towards the support, it must take down with it in this act all the bar be- 
tween it and the free end, just as if this were an index extending out from the 
point considered. The molecular actions by which this is effected will un- 
doubtedly give rise incidentally to small longitudinal strains, by which the 
relative positions of the molecules of the two contiguous sections will be some- 
what disturbed. Accordingly, the linear deflection of the end of the bar, re- 
sulting from the shearing stress taking effect along the whole length of the 
bar, should be proportional to the square of the length. The same conclusion 
will obviously apply to the case of a bar loaded at its middle and resting on 
two supports. The effect of the shearing stress should also be inversely pro- 
portional to the area of the cross section, or bd, and directly proportional to 

P P I 2 1 

— . We thus obtain, on theoretical grounds, the term C —. — . — . Now let 
a a 4 bd 

G = coefficient of elastic resistance to transverse shearing stress, and let m = 
-^, or O -— — . The value of G l for the distance 1 is -?$, or ~n. The new terra 

becomes, then, 5-= . -r-r-, for which we may take G —. 
o-hj od bd 

2 PI* 
11 The other term in the formula, viz., - . _, _.. differs from the ordinary ex- 

' '3 4:Ebd* y 

pression for the deflection due to the longitudinal strains on the fibres, in con 



DISTORTIONS. 235 



taining the factor ^. With regard to this I will only state here that this is 

o 

the expression for the deflection attendant on the longitudinal strains, which 
I deduced a year or two since, directly from the fundamental conception 
that these strains are the incidental effects of the shearing stress, without 
using the principle of the lever or of moments, I now find it to be verified 
by the results of my experiments. It may be well to add that this expres 
sion was not adopted, in my recent attempt to represent the experiments, 

until the ordinary expression j-prrj-i was f° un( i to ^ a ^> giving values of E far 
too large. 

" For the pine sticks the value of ra, as derived from the values of G and E 

by the relation C = 5^, falls between 2 and 4. It varies with the different 

sticks used, with the inclination of the layers to the horizontal. When the 
layers were horizontal, the value of m was 4 ; when they were vertical it was 
2. In one of the sticks the inclination was nearly 45°, and the value of m 
came out 2.93. In the case of another stick the inclination, in one experi- 
ment, was about 25°, and in another 65°; and the values of m obtained are 
2.18 and 3.55,- the average of which is 2.86. The values of i£ obtained varied 
with the different sticks from 933,000 to 1,093,000 lbs. 

"For the wrought-iron and steel bars (1 in. by \ in.) the value of m was a 
little less than 2 when the bar rested edgewise on its supports, and 4 when it 
rested flatwise ; or the same as with pine sticks when their layers were verti- 
cal or horizontal. This is a remarkable result, since it indicates that the bara 
were made up of laminse parallel to the breadth, with separating spaces of 
weaker molecular forces, called into action by vertical displacements. This 
condition of things is no doubt attributable to the operation of rolling, to 
which the bars have been subjected. This, I conceive, from theoretical con- 
siderations, ought to have had the effect to weaken the effective molecular 
forces in the vertical direction, and augment them in the horizontal direction. 
The value of E, for the iron bar, was found to be 25,220,000 lbs., and for the 
steel bar 25,333,000 lbs." 



182, DIRECTION OF MAXIMUM AND MINIMUM STRAINS 

at any point of the longitudinal section of a oeam. 
Let AB CD be an element, 

<p = the pull or push per unit of section AD, = — r—y 

Eq. (212). 
X= the shearing per unit on the surface AB, 



= W ^> E( i-( 213 ) 



236 



THE RESISTANCE OF MATERIALS. 



6%P 

which for rectangular beams is , 3 \_\d 2 — y*], Eq. (216), 

Z = the transverse shearing per unit on AD, = X, 

/3 = the variable angle ABD, 

_ZV~= the normal component of the stresses per unit on the 

plane BD, and 
Ss = the shearing stress per unit along BD. 









^ 



Fig. 113. 



For the sake of simplicity, consider a rectangular beam, 
which is fixed at one end and loaded with a weightP at the free 
end. Then %P = P, and we have 



p = 



12Pxy 

bd s ' 



6P 



We also have 



in 



(220) 



(221) 



X.A B — the total shearing on AB, 
Z.AD = X.AD — the total shearing on AD, 
p.AD = the direct pull or push on AD, 
N.BD = the total normal component on BD, and 
Ss.BD = the total shearing pull or push on BD. 

Resolving the forces normally and parallel to BD, and we 
have 

Ss.BD = AB. Xcos/3 - AD. Xsin/3 + AD.pcos/3 
JST.BD = AB.Xsm/3 + AD.Xco&/3 + ADpsinp 



DISTORTIONS. 237 

T3 , AB Q ,AD . Q 

~BD ~ C0S ^ ' an ~BD ~ Sm ^ 

,\ Ss = X(cos 2 /3 - sin 2 /3) +^sin/3cos/3 = Xcos2/3 + 

ij?sm20 (222) 

JV= 2Xsin/3cos/3 + ^sin 2 /3 = Xsin2/3 + Jj?(l - 

cos2/3) (223) 

Hence, Ss is a maximum for 

tang2/3 = fe = tangS/% (say) . ". (224) 

j) 2X 

••• sin2/?i = *frP ; cos2A = #n^ 

hence the maximum of the shearing force is 

£*m = V(ip) 2 + X 2 (224a) 

Similarly, for iVwe have for a maximum or minimum 

9 7" 

tang2ft = - — (225) 

.*. tang2ft tang2ft = — 1 

or 2ft = 90° + 2ft 
.'.A = 45°+ ft; 

hence the lines of maximum shearing cut the lines of maximum 

direct stress at angles of 45 degrees. 

2ft may hare two values a and 180° -fa .\/3 = -Ja or 90° + -Ja 

2X o 

sin2ft = =F ; cos2ft = ± , J , 

4/y + 4X 2 ' ^ Vp> + 4X 2 ' 

the upper signs correspond to a minimum and the lower to a 
maximum. 

The maximum value of iVis 

JV m = ip + 4/(ip) 2 + X 2 
and the minimum value is 

jsr = ip- 4/(^)2 + x 2 

On the compressive side of the beam we have 

Ss = X(cos 2 /3— sin 2 /?)— j9sin/3cos/3 
jar= 2Xsin/3cos/3 -^sin 2 /3 



238 the resistance of materials. 

Discussion. 

On the neutral axis p = 
.-. tang2ft = and 2ft =0or 180° .-. ft = or 90° 
tang2ft = oo ; 2ft = 90° or 270° /. ft = 45° or 135° 
and jSs m = Xand N m = Xand X = — X 

This shows that the intensity of the shearing stress on the 
neutral axis is a maximum along the axis, and is of the same 
value as at right angles to it and equals X The equality was 
shown in Art. 177. The maximum direct stress is normal to a 
section which is inclined 135 degrees to the axis, and the mini- 
mum at an angle of 45 degrees. Its value per unit is JV m = X 
This may be shown directly, for it is evidently the resultant 
of two rectangular shearing forces each equal to X, and hence 
is |/2X; but the area is V% times the horizontal unit ; hence 
the stress per unit will be 4/2X-T- \/2 — X, as given above. 

At the outer elements X= 

.-. tang2ft = oo ; 2 ft = 90° and ft = 45° 

tang2ft = — ; 2ft = 180° or 0° and ft = 90° or 0° 

X m =i?andi\7" o = 

That is, the maximum stress is normal to a section which is 
perpendicular to the neutral axis ; in other words, it is parallel 
to the axis and equals the pulling stress, as it should. The 
minimum value is zero in a direction normal to the surface, 
and the maximum shearing stress is along a section which is 
inclined 45 degrees to the axis, and its intensity is %p, which 
agrees with Article 177. 

For any point we have from Eqs. (220), (221), (224) and 
(225); 

tang2ft = - i d "-^ ......... (226) 

xy 

In the last- equation take successive values for x (as x — %l, 
f-Z, f I, etc.), and for each value substitute values of y (such as 
J- of \d> | of $d, etc.), and determine the corresponding values 



DISTORTIONS. 



239 



of /9o. Lay off the computed angles at the points whose coordi- 
nates are thus assumed. The lines of maximum direct stress will 
be normal to the lines thus constructed. But since the angles 
for a maximum and minimum differ by 90 degrees, the incli- 
nation of the lines so constructed will correspond in direction 
with (say) the maximum, and the normal to it with the mini- 
mum stress. By determining a sufficient number of points a 
network of lines may be drawn, as in Fig. 114, which represent 




Fig. 114. 

the direction of the lines of maximum and minimum stress, 
those concave downwards corresponding to tension, and those 
concave upwards to compression. 

The parts more nearly horizontal correspond to the maxi- 
mum, and the steeper parts to a minimum. They cross the 
neutral axis at angles of 45 degrees, and each other at all 
points at right angles, and the axes of minimum stress cut the 
surfaces at right angles, and the axes of maximum stress are 
parallel to it at the surface. 

In a similar manner lines of maximum shear may be drawn. 

To find the equation of one of these curves, we have (the 
axis of the stress being normal to the elementary section) 

-^ = cot/3 2 or -j- = tang/3 2 
ax ay 

But ta„g2 / 3 2 a J^lft = _ ^ 
1 — tang 2 /3 2 p 



•••fh *■*=+&* 



yi + 



j>< 



iX* 



*> or ^» 



240 THE RESISTANCE OF MATERIALS. 

whence, by Equations (220) and (221) 

\_\d?-f (i#-v) a J 

which is the differential equation of the curve, but I do not 
think that it can be integrated in finite terms. 

Remark. — The analysis by which Poisson and others determined that the 
coefficient of lateral contraction is J that of the longitudinal dilation, both 
per unit, has been criticised by Thomson and Tait. (See their Natural Phi- 
losophy, 1867, Vol. I., p. 521.) They give the following: — 

Let n = the rigidity, which according to our notation = G = \ (A — B), 

and k = the resistance to dilation = — = - (A + 2B) ; then the linear elon- 

1) o 

gation, i t = (— + ^ ) P = A z^ B _ 2B z P > and the linear contraction, 

'» = (fin ~ 9k) P = A 2 +AB-2B> P ' whi6h are the Valu6S ° f <l *** * 2 fo1 " 
lowing Eq. (209) in the pre .eding text. 
We have 

it B 



i x " A + B' 
in which, if the ratio is £, Swill be %A. These substituted in Eq. (208a) give 

as before found. That this result is approximately true for iron has been 
shown by the experiments of M. Wertheim.* 

For ordinary glass and crystal he found 2.4 nearly for the ratio. 

But there are some isotropic solids in which this is not the correct ratio, 
such as India rubber and elastic jellies. In such cases the value of B must be 
determined by experiment. 

1 assumed that = B = G, because it is approximately true for those solids 
which are more commonly used by the engineer, and also because it greatly 
simplifies the investigation. 

The problem of the distortion of a prism which is subjected to torsion has 
been thoroughly discussed by St. Venant. He determined the character of the 
sections which originally were plane and normal to the axis of torsion ; also 
determined the correction which should be applied to Coulomb's formula ; 
also compared his results with those of experiment, and deduced conclusions 
of great value to the engineer. This problem alone furnishes sufficient mate- 
rial to fill a volume. 

It was unnecessary to introduce the letter G into the notation on page 220 
and the following, since it is the same as E s previously used ; but I did so be- 
cause it has been used by other writers, and I desired to show its relation to 
my notation. 

Gomptes Rendus, t. xxviii., p. 126. 



EFFECT OF LONG-CONTINUED STRAINS. 241 



CHAPTER X. 

EFFECT OF LONG-CONTINUED STRAINS— OF OFT-REPEATED 
STRAINS, AND OF SHOCKS— REMARKS UPON THE CRYSTAL- 
LIZATION OF IRON. 

EFFECT OF LONG- CONTINUED STKALNS. 

183. general effect. — The values of the coefficients 
of elasticity and the modulii of tenacity, crushing, and of rup- 
ture were determined from strains which were continued for a 
short time — generally only a few minutes — or until equilibrium 
was apparently established ; and yet it is well known that if the 
strain is severe, the distortion, whether for extension, compres- 
sion, or bending, will increase for a long time ; and as for rup- 
ture, it always takes time to break a piece, however suddenly 
rupture may be produced. By sudden rupture we only mean 
that it is produced in a very short time. 

The increased elongation due to a prolonged duration of the 
strain beyond a few minutes will affect the coefficient of elas- 
ticity but very slightly, for the strains which are used in deter- 
mining it are always comparatively small, and the greater part 
of the effect is produced immediately after the stress is applied. 
If the distortion should go on indefinitely under the action of a 
constant load, no matter how slowly, the elasticity, and hence 
the coefficient, would be greatly modified by a very great dura- 
tion of the stress ; and at last rupture would take place. If the 
basis of this reasoning be well founded, we might reasonably 
fear the ultimate stability of all structures, and especially those 
in which there are members subjected to tension. But the con- 
tinued stability of structures which have stood for centuries, 
teaches us, practically at least, that in all cases in which the 
strain is not too severe, equilibrium becomes established between 
the stresses and strains, and in such cases the piece will sustain 
the stress for an indefinitely long time. 



242 



THE RESISTANCE OF MATERIALS. 



184. hodgkinson's experiotents. — The results of the 
experiments which are recorded in Article 45, page 52, show 
that in one case the compression increased with the duration of 
the strain for three-fourths of an hour. In the case of exten- 
sion on another bar, as shown in Article VII., page 7, it appears 
that the same weight produced an increased elongation for nine 
hours ; but during the last, or tenth hour, there was no increase 
over that at the end of the ninth hour. 

In both these cases the strain was more than one-half that of 
the ultimate strength. 

185. vicat»s experiments. — M. Yicat took wrought- 
iron wire and subjected it to an uniform stress for thirty-three 
months. The elongations produced by the several weights were 
measured soon after the weights were applied, and total lengths 
determined from time to time during the thirty-three months. 
It was found for all but the first wire, as given in the following 
table, that the increased elongations after the first one were 
very nearly proportional to the duration of the stress. {Annates 
de Chemie et Physique, Yol. 54, 2d series.) 

TABLE 

Of the Besults of M. Vicatfs Experiments on Wrought-iron Wire. 



Amount of Strain. 

\ of its ultimate tensile strength. 
\ of its -ultimate tensile strength. 
\ of its ultimate tensile strength. 
| of its ultimate tensile strength. 



<0 <D O 

-P -P « 

-2 ^ ^ 

88 £ » •* 



CD 



* * .2 § 

CD O e3 



Increased Elongation after 33 
months. 



No additional increase. 
0.027 of an inch per foot. 
0. 040 of an inch per foot. 
0.081 of an inch per foot. 



186. fairbaibn's experiments. — Fairbairn made ex- 
periments upon several bars of iron, which were subjected to a 
transverse strain, the results of some of which are recorded in 
the following tables. (See Oast and Wrought Iron, by Wm. 
Fairbaim). The bars were four feet six inches between the 



EFFECT OF LONG-CONTINUED STRAINS. 



243 



supports, and weights were applied at the middle, and permit' 
ted to remain there several years, as indicated by the tables. 
The deflections were noted from time to time, and the results 
were recorded. 

TABLE I. 

In which the Weight Applied was 336 pounds. 



Tempekatuee. 



78 c 
72 c 
61 c 
50 c 
58 c 



March 11, 1837.... 

June 3, 1838 

July 5, 1839 

June 6, 1840 

November 22, 1841. 
April 19, 1842 

Mean 



i* 


6 . 


c 8 




rS-S 


T3 O 


a 


I* 


■9 « 


s 9 


T3.2 


-£.3 


■3 +3 


M 


1.270 


1.461 


1.316 


1.538 


1.305 


1.533 


1.303 


1.520 


1.306 


1.620 


1.308 


1.620 


1.301 


1.548 






o§3> 

la! 
n 



Cold-blast, 
0.661 : 1 

Hot-blast, 
0.694 : 1 



Previous to taking the observations in November and April 
the hot-blast bar had been disturbed. 

In regard to this experiment Mr. Fairbairn remarks : — " The 
above experiments show a progressive increase in the deflec- 
tions of the cold-blast bar during a period of five years of 0.031 
of an inch, and of 0.087 of the hot-blast bar." The numerical 
results are found by comparing the first deflection with the 
mean of all the observed deflections. But an examination of 
the table shows that the greatest deflection, which was observed 
in both cases, was at the second observation, which was about 
a year and a quarter after the weight was applied, and during 
the next two years the deflections decreased 0.015 of an inch 
for the cold blast, and 0.018 of an inch for the hot-blast bar. 
After this the deflections appear to increase for the cold-blast 
bar 0.005 of an inch the next two years. Considering all the 



244: 



THE RESISTANCE OF MATERIALS. 



particulars of these experiments it does not seem just to con 
elude that the deflections would have gone on increasing indefi 
nitely with a continuance of the load. Admitting that the 
small increase of deflections during the last two years are cor 
rect and not due to errors of observation, we see no reason why 
the deflections would not be as likely to decrease after a time 
as they were after the first year. 



TABLE II. 
In which the Bar was Loaded with 392 pounds. 



TEMPEBATUKE. 


f 

o 

o . 

si 

A 


A 

■fl 

1-9 


ot-blast — deflec- 
tion in inches. 


'M 
StU 




D 


n 


M 




March 6, 1837.. . 


1.684 


1.715 




78° 


June 23, 1838 


1.824 


1.803 


For cold-blast, 


72° 


July 5, 1839 


1.824 


1.798 


0.771 : 1 


61° 


June 6, 1840 


1.825 


1.798 


For hot-blast, 


50° 


November 22, 1841. 


1.829 


1.804 


0.805 : 1 


58° 


April 19, 1842 


1.828 


1.812 






Mean 


1.802 


1.788 











Here we see a general increase in the deflections from year 
to year, but the changes are not entirely regular. The princi- 
pal increase is during the first year. In the cold-blast there 
was a slight decrease of deflection during the last year, and in 
the cold-blast it was less at the third and fourth measurement 
than at the second. 



EFFECT OF LONG-CONTINUED STRAINS. 
TABLE III. 



245 











M 






h 


S • 








Tempebatube. 


o 






1« 
p. H tao 






o . 


1-9 


*** 






a 
<s o 
a '-3 


■9 a 
S-2 


11 


SI 






ft 


o 


m 






March 6, 1837 


1.410 


§3 






78° 


June 23, 1838 


1.457 


■N 


Cold-blast, 




72° 


July 5, 1839 


1.446 






61° 


June 6, 1840 


1.445 


»s . 


0.881 : 1 




50° 


November 22, 1841. 


1.449 


O'S'S 






58° 


April 19, 1842 

Mean 


1.449 










1.442 















We find from this table, as from Table L, that the maximum 
deflection was observed about a year and a quarter after the 
weight was applied, and that it decreased during the next two 
years, after which it slightly increased. The deflections were 
the same at the two last observations. These changes took 
place under the severe strain of more than four-fifths of the 
breaking weight. These experiments indicate that for a steady 
strain which is less than three-fourths of the ultimate strength 
of the bar, the deflection will not increase progressively until 
rupture takes place, but will be confined within small limits. 

187. KOEBiiiNcs observations. — The old Monongahela 
bridge in Pennsylvania, after thirty years of severe service, was 
removed to make place for a new structure. The iron which 
was taken from the old structure was carefully examined and 
tested by Mr. Roebling, and found to be in such good condition 
that it was introduced by him into the new bridge.* 

lie also found that the iron in another bridge over the Alle- 



* Roebling's Report on the Niagara Railroad 
Frank. Inst., 1860, Vol. LXX, p. 361. 



Bridge, 1860, p. 17 ; Jour. 



THE RESISTANCE OF MATERIALS. 

ghany Eiver was in good condition after forty-one years ot 

service. 

188. oft-repeated strains. — Nearly all kinds of struc 
tures are subjected to greater strains at certain times than at 
others, and some structures, as bridges and certain machines, 
are subject to almost constant changes in the strains. Loads 
are put on and removed, and the operation constantly repeated. 
The following experiments for determining the effect of a load 
which is placed upon a bar and then removed, and the operation 
of which was frequently repeated, were made by Wm. Fair- 
bairn, in I860.* The beam was supported at its ends, and the 
weight which produced the strain was raised and lowered by 
means of a crank and pitman, as in Fig. 115. 




The gearing was connected with a water-wheel, which was 
kept in motion day and night, and the number of changes of 
the load were registered by an automatic counter. The beam 
was 20 feet clear span and 16 inches deep. The dimensions of 
the cross section were as follows : 

Top — Plate, 4 x \ = 2.00 sq. inches. 

Angle irons, 2x2x^ = .. .2.30 " " 

Bottom —Plate, 4x^= 2.00 " " 

Angle irons, 2 x 2 x ^=.1.40 " " 
Web— Plate, 15J x i = . . .■ 1.90 " " 

Total 8.60 " « 

Weight of beam, 1 cwt. 3 qrs. 3 lbs. 
Probable breaking weight, 9.6 tons. 

* Civ. Eng. and Arch. Jour., Vol. XXIII., p. 257, and Vol. XXIV., p. ?B7. 



EEFECT OF OFT-BEPEATED STBATNS. 



241 



First Experiment. — Beam loaded to \ the breaking weight :— 

Total applied load '. 5,809 lbs. 

Half the weight of the beam 434 " 

Strain on the bottom flange 4.3 tons per sq. inch. 

Margin of strength by Board of 
Trade 3.4 

TABLE 

Of the Results of Experiments made upon a Beam which was Supported at its 
Ends, and a Weight repeatedly but gradually Applied at the Middle. 



DATE. 


OS 

9 

§ 

O 

<w 
o 

d 


| 

a 
o 

o 

<c o 
P 


DATE. 


! 

■s 

1 


! 

o 

si 

•!| 
1*8 



ft 


I860. 
March 21 


10,540 

15,610 

27,840 

46,100 

57,790 

72,440 

85.960 

97.420 

112.810 

144,350 

165,710 

202.890 

235.811 


0.17 

0.18 
0.16 

0.16 
0.17 
0.17 
0.17 
0.17 
0.17 
0.16 
0.18 
0.17 
0.17 


1860. 

April 13 


268,328 
281,210 
321,015 
343,880 
390,430 
408,264 
417,940 
449,280 
468,600 
489,769 
512,181 
536,355 
560,529 
596,790 


0.17 


22 


14 


0.17 


23. 


17 


017 


24 


20 


0.17 


26 


25 


0.16 


27...- 


27 


0.16 


28 


28 


0.16 


29 


May 1 


0.16 


30 


3 


0.16 


31 


5 


0.16 


April 2 


7 


0.16 


4 


9 


0.16 


7 


11.. 


016 


10 


14 


0.16 









At this point, after half a million of changes, the beam did 
not appear to be damaged. At first it took a permanent set of 
0.01 of an inch, which did not appear to increase afterwards, 
and the mean deflection for the last changes were less than for 
the first. For the last seventeen days the deflection was uni- 
form, but for the first seventeen days it was variable. 

The moving load was now increased to one-third the break- 
ing weight, = 7,406 lbs., with the following results : 



248 



THE RESISTANCE OF MATERIALS. 



DATE. 


S 

Xt 

o . 


I 
a 

3 

s 

o 
33 

S 
« 
5 
P 


DATE. 


o 

1 

S3 
c3 
Xi 

O . 

r 


£ 

i 

a 

c 

a 
o 

p 


1860. 
May 14 


12,623 

36,417 
53,770 
85,820 
128,300 
161,500 
177,000 
194,500 


0.22 
0.22 
0.22 
0.21 
0.22 
0.22 
0.22 
0.22 
0.21 


1860. 
June 7 


217,300 
236,460 
264,220 
292,600 
327,000 
350,000 
375,650 
403,210 


0.21 


15 


9 


0.21 


17 


12 


0.21 


19 


16 


0.22 


22 .. 


21 


0.23 


26 


23 


0.25 


29 


25 


0.23 


31 


26 


0.23 


June 4 











The beam had now received 1,000,000 changes of the load, 
but it remained uninjured. The moving load was now in- 
creased to 10,050 lbs. — or one-half the breaking- weight — and it 
broke with 5,175 changes. The beam was then repaired by 
riveting a piece on the lower flange, so that the sectional area 
was the same as before, and the experiment was continued. 
One hundred and fifty-eight changes were made with a load 
equal to one-half the breaking weight ; and the load was then 
reduced to two-fifths the breaking weight, and 25,900 changes 
made. Lastly, the load was reduced to one third the breaking 
weight, with the following results : — 



EFFECT OF LONG-CONTINUED STRAINS. 



249 



DATE. 


•s 

5 

! 

a . 


a 

o 
c 
t-t 

c 

1 

a) 
IS 

P 


! 

DATE. 


"8 

3 

-a 
o . 


m 

1 

a 
a 

I 

P 


1860. 
August 13 


25,900 

46,326 
71,000 
101,760 
107,000 
135,260 
140,500 
189,500 
242,860 
277,000 
320,000 
375,000 
429,000 
484,000 
538.000 
577,800 
617,800 
657,500 
712,300 
768,100 
821,970 
875,000 


0.1* 


1860. 
1 Dec. 22 


929,470 
1,024,500 

1,121,100 
1,278.000 
1,342.800 
1,426,000 
1,485,000 
1,543,000 
1,602,000 
1,661,000 
1,720,000 
1,779,000 
1,829,000 
1,885,000 
1,945,000 
2,000,000 
2,059,000 
2,110,000 
2,165,000 
2,250,000 
2,727,754 


18 


16 


29 




20 

24 


1861. 
Jan. 9 




25 


19 




31 


26 


Sept. 1 


Feb. 2 


8 


11 


15 


16 


22 


23 




30 


March2 




October 6 


9 


18 


13....... 


13 


17 


20 


23 




27 


30 




November 3 


April 6 


10 


13 


17 


20 


23 


27 


December 1 


May 4 


8 


11.:..::::... 


15 


5 June 


0.17 



The piece had now received nearly 4,000,000 changes in all, 
but the 2,727,000 changes after it was once broken and re- 
paired did not injure it. The changes were not very rapid. 
During the first experiment they averaged about 11,000 per 
day, or less than eight per minute, and during the last experi- 
ment the highest rate of change appears to have been less than 
eleven per minute, which is very slow compared with the 
strokes of some forge hammers. 

189. stiffening un»er strain. — The experiments re- 
corded in Articles 31, 185, and 186, indicate that iron may be- 
come stiff er, if not stronger, under strain. That such is the 
fact has recently been confirmed in a very striking manner by 
the experiments of Professor Thurston (hereafter given), and 
the following experiment, which was made by Commander L. 
W. Beardslee, of the U. S. Navy.* 

* Reported to the Am. Soc. of Civ. Eng., New York, Nov., 1874. The 
specimen is preserved in the Stevens Institute of Technology. 



250 THE RESISTANCE OF MATERIALS. 

The bar was of Phoenix iron, with an eve formed at each 
end. It was pulled apart by a hydraulic machine, and registered 




Fig. 116. 



with weights and levers. It was originally If inches in diame- 
ter, but was drawn clown to If of an inch at a, when it suddenly 
broke in the weld near the eye, with 67,800 pounds. The rest 
of the bar was slightly tapered, but with no marked diminution. 

A new eye was welded on, and the next day it was put in the 
testing machine, when, instead of breaking at a, it began to 
yield and draw down at b, and finally broke at that point under 
a strain of 88,000 pounds. 

Such are the facts in regard to this remarkable fracture. 
Had the eye not broken, it is quite certain that the bar would 
have broken at a. The particles then were moving (flowing) 
over each other more rapidly than at any other point, and were, 
apparently, on the verge of separation. By being relieved of 
the strain for a day (resting) its strength was greatly increased, 
so that it was stronger at the reduced section on the second day 
than the full section was on the former clay. It would appa- 
rently have broken on the first day with a strain somewhat ex- 
ceeding 68,000 pounds, but on the second day it sustained 
88,000 pounds at that point without fracture, which is a very 
large increase in the strength. 

The most that can be said with certainty is, that the particles 
by flowing over each other, and having time to come to rest in 
their new positions, the cohesive force between them was in- 
creased. The contraction necessarily develops heat, and it is 
probable that the heat during the flowing, and the abstraction 
of it afterwards, played an important part in securing the in- 
creased cohesion. 

190. shocks. — In a broad sense, a shock is the impinging 
of one mass against another,- whereby the velocity of one or 
both of the masses is suddenly changed. In common language 



EFFECT OF SHOCKS. 251 

it is a blow produced by one solid body striking another. In 
the impact of gases, liquids and semi-fluids, shocks are not con- 
sidered. When the motion of a rigid body is gradually changed 
like the connecting rod, or the pitman of an engine, shocks are 
not produced. If a moving mass be brought to rest by a resist- 
ance acting through a finite space, the shock is much less than 
if it be arrested more suddenly. Thus a forge hammer in 
striking a molten mass of metal produces but little shock, whereas 
the same blow upon cold metal may produce a severe shock, 
"No moving mass can be brought to rest instantaneously ', but 
the more rigid the masses, and the more unyielding the supports 
of the body receiving the blow, the more suddenly will the 
blow be arrested, and the more severe will be the shock. 

The effect of shocks may be greatly modified by the intro- 
duction of springs. Thus, the use of steel, rubber and wooden 
springs in vehicles and machines are familiar examples, and if 
the springs have but little mass, and have sufficient range of 
action, they may very nearly remove the effect which shocks 
would otherwise produce. 

Oft-continued and long-repeated shocks upon metals are 
quite certain to produce fracture sooner or later. One who is 
unaccustomed to these effects is apt to be surprised at the fail- 
ure of iron or steel after it has sustained a moderate shock for 
a long time, but those who are accustomed to them seek to an- 
ticipate and provide against them. All metals in use have their 
"life." In some cases they are worn out, but in many others 
they break after a time. They can sustain only a certain 
amount of service. All machinery, tools, implements, vehicles, 
etc., have to be renewed. But there is nothing more uniformly 
disastrous to machinery, or which produces results more unex- 
pectedly than shocks. 

The following example is a good illustration of its effects. 

To aid in the handling of large masses of iron while being 
forged, a long bar of iron is sometimes forged to them to serve 
as a handle. This handle is called a u Porter bar," and may be 
used repeatedly for the same purpose. 

At the West Point Foundry a Porter bar, which had been in 
use about twenty years, broke near the middle whilst the ham- 
mer was at work upon the forging which was attached to the 



252 THE RESISTANCE OF MATERIALS. 

other end. The bar was about twenty feet long and twelve 
inches in diameter at the smaller end, and twenty-three inches 
at the end where it was attached to the forging. It was about 
fourteen inches in diameter where it broke. It was slung on a 
chain in the usual manner, and the fracture was between the 
free end and where it was slung, and some two or three feet 
from the latter place. 

The appearance of the fracture was described as highly crys- 
talline and a clean break. The piece broken off probably 
weighed a ton and a half. It would have required a load of 
nearly fourteen tons applied at the end to have broken it if the 
iron was sound. 

This is a remarkable fracture for iron. It is not probable 
that the iron was crystallized, but that it had that (vppea/rcmoe 
on account of the character of the breakage, as will be explained 
hereafter. 

The heavy end, which served as a handle, was caused to 
vibrate under the action of the hammer, and doubtless caused 
excessive strains which started a fracture ; and by repeating 
the operation from time to time finally caused rupture. 

The writer is familiar with similar examples in the case of 
steel. Where the steel had been subjected to repeated shocks, 
one end of a bar would drop off while the smith was at work 
upon the other end. 

The fracture in such cases is doubtless a slow process. At 
first a mere crack is started, which increases slowly by the re- 
peated blows, but is unseen by the observer until the piece is so 
much weakened that it fails suddenly at last. 

The effect of a low temperature upon metals when subjected 
to shocks is not fully determined. "When subjected to a steady 
tensile strain, numerous experiments prove conclusively that 
iron is stronger at very low temperatures than at ordinary tem- 
peratures. But it is commonly supposed that machinery, tools, 
rails on the railroad, tyres on locomotives, axles under the cars, 
etc., break more easily when cold than -when warm. Steel rails 
when they first came into use were supposed to be more liable 
to break when cold than iron ones, but they have now- come 
into extensive use, and there are no more breakages than for- 
merly, and probably not as many. 



EFFECT OF SHOCKS. 253 

Mr. Sandberg, the translator of Styffe's work,* thought it 
probable that iron when subjected to shocks might not give the 
same relative strength at different temperatures that it would 
when subjected to a steady strain. He therefore instituted a 
series of experiments to satisfy himself upon this important 
point, and aid in solving the problem. The following is an 
abstract of his report : — 

The supports for the rails in the experiments were two large 
granite blocks which rested upon granite rocks in their native 
bed. The rails were supported near their ends on these blocks. 
They were broken by a ball which weighed 9 cwt., which was 
permitted to fall five feet the first blow, and the height increased 
one foot at each succeeding fall, and the deflection measured 
after each impact. A small piece of wrought-iron was placed 
on the top of each rail to receive the blow, so as to concentrate 
its effect. 

The rail was thus broken into two halves, and each part was 
afterward broken at different temperatures. As the experi- 
ments were not made till the latter part of the winter, the 
lowest temperature secured was only 10° Fahr. Fourteen rails 
were tested : — Seven of which were from Wales ; five from 
France ; and two from Belgium. From these the experimenter 
drew the following conclusions : — 

1. " That for such iron as is usually employed for rails in the 
three principal rail-making countries (Wales, France, and Bel- 
gium), the breaking strain, as tested by sudden blows or shocks ^ 
is considerably influenced by cold ; such iron exhibiting at 10° 
F., only one-third to one-fourth of the strength which it pos- 
sesses at 84° F. 

2. " That the ductility and flexibility of such iron is also 
much affected by cold, rails broken at 10° F., showing on an 
average a permanent deflection of less than one inch, whilst the 
other halves of the same rails, broken at 84° F., showed less 
than four inches before facture." 

These experiments seem to be conclusive fo~ the iron which 
was tested. 



* The Elasticity, Extensibility, and Tensile Strength of Iron and Steel By 
Knut Styffe. Translated by Christer P. Sandberg, Londcn. 



254 THE RESISTANCE OF MATERIALS. 

From the official reports of the Verein DeutscAer Eisenbahn 
Verwaltungen, it appears that during the year 1870, on 22 lines 
belonging to the Association, 132 axles of locomotives, tenders, 
and carriages were broken. In comparison with the previous 
year, in which the fractures amounted to 163, these figures show 
an improvement. There is a decrease of 19.3 per cent, on the 
service which, considering the extraordinary demands occa- 
sioned by the Franco-German war, and the increase of rolling 
stock in Austria, appears considerable. The fractures either 
occurred or were reported in the months of 

December, January, February, in 39 cases. 
March to May (inclusive), in 30 " 

June to August, " in 25 " 

September to November " in 38 " 

The influence of the cold season, despite much that has recently 
been said to the contrary, is distinctly marked ; from March 
till August 55 only, and during the other months 77 axles 
broke. The average run of the axles broken in 1870 was as 
follows : — 

Locomotives 11 years 4 months 13 days. 

Tenders, .13 " 4 " 20 " 

Carriages 11 "11 " 13 " 

Average 12 " 2 " 29 " 

The average mileages of the axles were in the case of 

Locomotives 34,241.7 miles (German). 
Tenders.... 31,494.5 " " 

Carriages... 24,040.1 " " 

Average. . . . 27,631.1 " 



a 



The maximum mileage attained was 69,000 miles. 

But in opposition to this we have the Report of the Massa- 
chusetts Railroad Commissioners for 1874. On page 74 of this 
report are the following conclusions : — " Cold does not make 
iron or steel brittle, or unreliable for mechanical purposes." 
" It is not the rule that the most breakages occur on the coldest 
days." " The introduction of steel, in place of iron rails, has 
caused an almost complete cessation of the breakage of rails." 



EFFECT OF SHOCKS. 255 

This report, which is the latest upon this point, shows that 
there must have been a great improvement in some respects in 
order to secure it. 

These results are in opposition to previously formed ideas in 
regard to the effect of cold. It being the latest report, and 
from a reliable source, we must look for an explanation in the 
improved character of the materials in the rails or in the sub- 
structure. There doubtless remains much to be learned upon 
this subject. It is especially desirable to determine the effect 
of the imparities in the metal. It is probable that those ele- 
ments which make iron cold-sheet will cause it to be more 
brittle at low than at moderate temperatures ; and that good 
metals will resist shocks better at low temperatures than at 
moderate ones. 

The following experiments, by John A. Roebling,* bear upon 
this subject : 

" The samples tested were about one foot long, and were re- 
duced at the centre to exactly three-fourths of an inch square, 
and their ends left larger, were welded to heavy eyes, making 
in all a bar three feet long. These were covered with snow 
and ice, and left exposed several da} T s and nights. Early in the 
morning, before the air grew warmer, a sample inclosed in ice 
was put into the testing-machine and at once subjected to a 
strain of 26,000 pounds, the bar being in a vertical position, 
and left free all around. The iron was capable of resisting 
70,000 lbs. to 80,000 lbs. per square inch. A stout mill-hand 
struck the reduced section of the piece, horizontally, as hard as 
he could, with a billet one and a half inches in diameter and 
two feet long. The samples resisted from three to one hun- 
dred and twenty blows. With a tension of 20,000 lbs. some 
good samples resisted 300 blows before breaking." 

The finest and best qualities of iron, or those that have the 
highest coefficient elasticity will resist vibration best. Tt is 
generally supposed that good iron will resist concussions much 
better than steel. Sir William Armstrong, of England, says : — ■ 
" The conclusion at which I have long since arrived, and which 
I still maintain, is, that although steel has much greater tensile 

* Jour. Frank. Inst., vol. xl., 3d series, p. 361. 



256 THE KESISTANCE OF MATERIALS. 

strength than wrought iron, it is not as well adapted to resist 
concussive strains." This was written many years since, but 
at the present day many mechanics prefer iron to steel for 
resisting shocks. 



191. crystallization of iron It is observed that 

metals which are subjected to oft-repeated and long-continued 
shocks become weak ; and when broken in this way they ap- 
pear to be crystallized, having apparently undergone a change 
of structure. A crystal is a homogeneous inorganic solid, 
bounded by plane surfaces, systematically arranged. The 
quartz crystal is a familiar example. Different substances 
crystallize in forms which are peculiar to themselves. Metals, 
under certain circumstances, crystallize ; and if they are broken 
when in this condition the fracture shows small plane sur- 
faces, which are the faces of the crystals. It is found in all 
cases that crystallized iron is weaker than the same metal in 
its ordinary state. Ey its ordinary state we mean that wrought 
iron is fibrous, and cast iron and steel are granular in their 
appearance. 

Iron crystallizes in the cubical system.* Wholer, in break- 
ing cast-iron plates readily obtained cubes when the iron had 
long been exposed to a white heat in the brickwork of an iron 
smelting furnace. 

Augustine found cubes in the fractured surface of gun bar- 
rels which had long been in use. 

Percy found on the surface and interior of a bar of iron, 
which had been exposed for a considerable time in a pot of 
glass-making furnace, large skeleton octahedra. (He seems to 
differ from the preceding in regard to the form of crystals.) 

Prof. Miller, of Cambridge, found Bessemer iron to consist 
of an aggregation of cubes. 

Mallet says : — " The plans of crystallization group themselves 
perpendicular to the external surfaces." 

Bar iron will become crystalline if it is exposed for a long 
time to a heat considerably below fusion. Hence we see why 
large masses which are to be forged may become crystalline, 

* Osborn's Metallurgy, pp. 83 >86. See Appendix. 



CRYSTALLIZATION OF IRON. 257 

on account of the long time it takes to heat the mass. Forging 
does not destroy the crystals, and forging iron at too low a 
temperature makes it tender, while steel at too high a tempera- 
ture is brittle. The presence of phosphorus facilitates crystal- 
lization. 

Time, in the process of breaking iron, will often determine 
the character of the fracture. If the fracture is slow, the iron 
will generally appear fibrous ; but if it be quick, it may appear 
more or less crystalline. This result has been frequently no- 
ticed. At Shoeburyness armor-plates were shattered like glass 
under the impact of shot at a velocity of 1,200 feet to 1,600 
feet per second. They were were made of good fibrous iron. 

William Fairbairn says: — *"¥e know that in some cases 
wrought iron subjected to continuous vibration assumes a crys- 
talline structure, and that then the cohesive powers are much 
deteriorated ; but we are ignorant of the causes of this change." 

The late Robert Stephenson f stated that in all the cases 
investigated by him of supposed change of texture, he knew 
of no single instance where the reasoning was not defective in 
some important link. 

Mr. Brunei accepted the theory of molecular change, for a 
time, as due to shocks, but afterwards expressed great doubts 
as to its correctness, and thought that the appearance depended 
more upon the manner of breaking the metal than upon any 
molecular change. 

Fairbairn presented his view of the probable cause of the 
internal change when it takes place in his evidence before the 
Commissioners appointed to inquire into the application of 
iron to railway structure. He says : — u As regards iron it is 
evident that the application and abstraction of heat operates 
more powerfully in effecting these changes than probably any 
other agency ; and I am inclined to think that we attribute too 
much influence to percussion and vibration, and neglect more 
obvious causes which are frequently in operation to produce 
the change. For example, if we take a bar of iron and heat it 
red hot, and then plunge it into water, it is at once converted 



* Civ. Eng. and Arch. Jour., Vol. iii., p. 257. 
f Am. R Times, March 6, 1869, Boston. 
IT 



258 THE RESISTANCE OF MATERIALS. 

into a crystallized instead of a fibrous body ; and by repeating 
this process a few times, any description of malleable iron may 
be changed from a fibrous to a crystalline structure. Vibration, 
when produced by the blows of a hammer or similar causes, 
such as the percussive action upon railway axles, I am willing 
to admit is considerable ; but I am not prepared to accede to 
the almost universal opinion that granulation is produced by 
those causes only. I am inclined to think that the injury done 
to the body is produced by the weight of the blow, and not by 
the vibration caused by it. If we beat a bar with a small ham- 
mer, little or no effect is produced ; but the blows of a heavy 
one, which will shake the piece to the centre, will probably give 
the key to the cause which renders it brittle, but probably not 
that which causes crystallization. The fact is, in my opinion, 
we cannot change a body composed of a fibrous texture to that 
of a crystalline character by a mechanical process, except only 
in those cases where percussion is carried to the extent of pro- 
ducing considerable increase of temperature. We may, how- 
ever, shorten the fibres by continual bending, and thus render 
the parts brittle, but certainly not change the parts which were 
originally fibrous into crystals. 

" For example, take the axle of a car or locomotive engine, 
which, when heavily loaded and moving with a high velocity, 
is severely shocked at every slight inequality of the rails. If, 
under these circumstances, the axle bends — however slightly — 
it is evident that if this bending be continued through many 
thousand changes, time only will determine when it will break. 
Could we, however, suppose the axle so infinitely rigid as to 
resist the effects of percussion, it would then follow that the 
internal structure of the iron will not be injured, nor could the 
assumed process of crystallization take place." 

The late John A. Roebling, who designed and constructed 
the Niagara Railway Suspension Bridge, in his report on that 
structure in I860,* says he has given attention to this subject 
for years, and as the result of his observation, study and experi- 
ment, gives as his view that " a molecular change, or so-called 
granulation or crystallization, in consequence of vibration or 

* Tour. Frank. Inst., Vol. xl., 3d series, p. 361. 



CRYSTALLIZATION OF IRON. 250 

tension, or both combined, has in no instance been satisfactorily 
proved or demonstrated by experiment." " I further insist 
that crystallization in iron or any other metal can never take 
place in a cold state. To form crystals at all, the metal must 
be in a highly-heated or nearly molten state." But he states 
that he is witnessing the fact daily that vibration and tension 
combined will greatly affect the strength of iron without chang- 
ing its fibrous texture. 

In speaking of the rock-drilling engines used in Hoosac 
Tunnel, Mass., which were driven by compressed air, the com- 
mittee says : * — " Gradually they began to fail in strength ; 
the incessant and rapid blows — counted by millions — to which 
they are subjected, appearing to granulate or disintegrate por- 
tions of the metals composing them." 

In some recent experiments made in France, interesting in- 
formation has been made known in regard to crystalline struc- 
ture in wrought-iron. The apparatus consisted of a bent axle, 
which was firmly fixed up at the elbow in timber, and which 
was subjected to torsion or twist by means of a cog-wheel con- 
nected at the end of the horizontal part. At each turn the 
angle of torsion was 24 degrees, and a shock was produced each 
time that the bar left one tooth to be raised by the next. Seven 
axles were submitted to the trial. In the first the movement 
lasted one hour, 10,800 revolutions and 32,40.0 shocks being pro- 
duced ; the axle, 2^- inch diameter, was taken from the machine 
and broken by an hydraulic press, but no change in its texture 
had occurred. In the second, a new axle having been tried 
4 hours, sustained 129,000 torsions, and was afterwards broken 
by means of an hydraulic press ; no alteration was perceptible 
to the naked eye, but, tried by a microscope, the fibres appeared 
without adhesion, like a bundle of needles. A third axle was 
subjected during 12 hours to 338,000 torsions, and broken in 
two ; a change in its texture and an increased size in the grain 
of the iron were observed by the naked eye. In the fourth, 
also, the axle was broken in many places after 110 hours and 
2,553,000 torsions. In the fifth, an axle submitted to 23,328,000 
torsions during 720 hours, was completely changed in its texture. 

* Annual Report of the Commissioners on the Troy and Greenfield Railroai 
and Hoosac Tunnel. House Doc., No. 30, p. 5, Boston, Mass. 



260 THE RESISTANCE vF MATERIALS. 

In the sixth, after ten months, during which the axle was sub- 
mitted to 78,735,000 torsions and shocks, fracture produced by 
an hydraulic press showed clearly an absolute transformation, 
the surface of the rupture being scaly, like pewter. The sev- 
enth axle, submitted to 129,304,900 torsions, presented a sur- 
face of rupture like that of the sixth, the crystals were found to 
be perfectly well-defined, it having lost every appearance of 
wrought iron. — IT. S. R. li. mid Mining Register, 1872. 

The last experiment looks like a proof of the fact that the 
internal structure of iron may be changed by strains and shocks, 
but in this example millions of them did not produce rupture. 
Much depends upon the severity of the shock. The rapidity 
of the blows also has much to do with its durability, since a 
rapid movement of the particles may develop heat to such an 
extent as to become an important element in the effect produced 
upon the metal. 

These several facts, though apparently somewhat conflicting, 
show quite conclusively, that some metals will crystallize under 
certain conditions ; that under certain conditions they may be 
strained millions of times without being damaged, or at least 
without being broken ; that under certain conditions strains and 
shocks combined may produce crystallization ; that shocks when 
severe will weaken metals, and if they are sufficiently numerous, 
will produce rupture. Much evidently remains to be learned 
upon this subject. There is a metal called " Phospho-Bronze," 
which combines in a remarkable degree toughness, rigidity, 
hardness and great elastic resistance, which, it is said, will not 
crystallize under repeated strains or continued vibration. 

193. the practical question is, how shall the life of 
such machines as are necessarily subjected to shocks, be pro- 
longed. The steam forge hammer (Nasmyth's) has been very 
troublesome on account of its frequent breakages. The auto- 
matic valve arrangement was so troublesome that many preferred 
to work them entirely by hand, but at the present day there are 
many in which this is as durable as any other part of the ma- 
chine. One of the essential features is to take the motion off 
from the hammer or piston-rod by a slope, so that the move- 
ment of the valve and its mechanism will be gradual. The 



CRYSTALLIZATION OF IKON. 



261 



piston-rod is liable to break. In some very heavy hammers the 
rod is keyed to the block and the end, which is square, presses 
against blocks of wood which are put in place for the purpose 
of relieving the shock, but this only partially cures the evil, 
since the rod is liable to break at the keyhole. 

A Mr. Webb, of England, proposed the improvement shown 
in Fig. 117. 




Fig. 117. 



Referring to the figure, it will be seen that the piston-rod, 
which is for the main part of its length 4 in. in diameter, is en- 
larged at the lower end to 6J in. in diameter, and is shaped 
spherically. This spherical portion of the rod is embraced by 
the annealed steel castings, B B, which are secured in their 
place in the hammer-head by the cotters, A, and the whole 
thus forms a kind of ball-and-socket joint, which permits the 
hammer head to swivel slightly on the rod without straining 
the latter. Mr. Webb first applied this form of hammer-rod 
fastening to a five-ton Nasmyth hammer with a 4 in. rod. With 
the old mode of attachment, with a cheese end, this hammer 
broke a rod every three or four weeks when working steel, while 
a rod with the ball-and-socket joint, which was put in in No- 
vember, 1867, has been working ever since, that is, to some 
time in 1869, without giving any trouble. The inventor hae 
also applied a rod thus fitted to five-ton Thwaites and Carbutt's 
hammer with equal success. 




262 THE RESISTANCE OF MATERIALS. 

One Morrison avoided the difficulty by making the rod very 
large and of uniform size, from the piston down to the hammer 
face. 

Mr. Samuel Trethewey, of Pittsburgh, Pa., thought, by re- 
ducing the rod at A, that he would compel the breakage tc 
take place at that point, and that the repairs could be 
more quickly made than when the breakage was 
permitted to take place at any point ; but, to his sur 
prise and gratification, the rods lasted from two to 
three times as long without breaking as they did 
when the rods were of uniform size. The ends 
taper one-half inch to twelve inches. 

Steam rock-drilling machines are of more recent 
date than steam hammers, but partake of the same 
difficulties, and many more in addition to them. 
They must be portable, and hence comparatively 
Fig. 118. light ; but they have severe work to do, and hence 
should be very strong. But for durability they 
must have mass. 

One of the ways of making such machines successful is to 
learn by practical experience where they are liable to break, 
and provide them with duplicate parts. 

Another efficient way of improving them is to make some 
simple non-expensive part, such as a bolt, pin, rod or bar, com- 
paratively weak, so that it will break first. The main parts 
will thus be preserved, and an ordinary mechanic may make 
the repairs. The use of a wooden pin for connecting the parts 
which would break when the machine met with a serious ob- 
struction, has greatly prolonged the life of certain machines. 
All reenterent sharp angles should be avoided in machines sub- 
jected to shocks. 

The cause of breakages has sometimes been attributed to 
crystallization, when the true cause was a lack of strength. In 
case of percussive forces the strain may exceed the amount 
estimated, and thus damage the material. As a general rule in 
such cases, the greater the amount of metal in the structure, 
when properly proportioned, the longer will be its life. 

The life of metals, or the amount which they will endure in 
performing a certain duty, is being determined approximate!} 



EFFECT OF LONG-CONTINUED STRAINS. 



263 



by actual use. Having determined it, other specimens of simi- 
lar quality, when used for the same purpose, should be cast 
aside before they fail, after having performed nearly the same 
duty. 

193. thurston's experiments.* — Professor Thurston, 
of the Stevens Institute of Technology, has made an entensive 
series of experiments upon various materials with a machine of 
his own invention, the prominent feature of which is its auto- 
matic registry. For the sake of simplicity, compactness, and 
economy, he so constructed the machine as to subject the 





Fig. 119. 



Fig. 120. 



specimens to torsion. It records automatically at every instant 
the moment of the stress, and the total angle of torsion. This 
feature enables one to make experiments rapidly and accu- 



* Papers read before the Am. Civ. Eng. Society, IS". Y., Jour, of the Frank, 
Inst. 1874. 



264 THE .RESISTANCE OF MATERIALS. 

rately, and by means of it many qualities may be detected 
which otherwise might escape observation. 

The twisting force is applied at F, Fig. 119, and the resist- 
ance is offered by a weight JD. The arms B and O turn upon 
independent axes, an enlarged view of one of which is shown 
in Fig. 120. The end U swings in the frame, while the other 
end is free. At the free end is a rectangular recess If for 
receiving one end of the specimen, which is usually made of the 
form shown in Fig. 121. The reduced part is one inch long and 
five-eighths of an inch in diameter. The other axis faces thisj 
but a short space is left between the free ends at M. 




Fig. 121. 

When the specimen is secured in the rectangular recesses, 
the axes are virtually connected by the specimen, so that as a 
force is applied to the arm F, tending to turn it on its axis, it 
will at the same time tend to turn the arm B on its axis ; but 
as the weight JD is moved from the vertical position it will 
bring a torsive strain on the specimen, and the farther it is 
forced out the greater will be the strain. The statical moment 
of the weight JD will equal the moment of the torsional stress. 
The relative angular movements of the arms C and B will be 
the measure of the total angle of torsion. It is evident that as 
the specimen yields to the strain, the arm C must travel farther 
than the arm B, in producing a given strain. % 

A guide curve F, of such form that its ordinates are propor- 
tional to the torsional moments, and its abscissas proportional 
to the arcs moved over by the arm B, is attached to the frame 
AA f . The other arm C carries a cylinder G, upon which 
paper is clamped for receiving the record. A pencil is secured 
to the arm B in such a way that it will be carried around with 
it, but which, at the same time, is free to move outward as it 
is moved along the curve F. 

After the specimen paper and pencil are arranged, the arm 
C is forced around, and arm B is thus forced forward, and the 



EFFECT OF LONG-CONTINUED STRAINS. 265 

pencil describes a line upon the paper as they move. The 
abscissa of the line will represent the angle of torsion and the 
ordinates the moment of stress. 

The interpretation of the diagrams has been the subject of 
much study. The comparison of the diagrams of a variety of 
materials, and a knowledge of the properties of some of them, 
enables the investigator to draw many general conclusions. 
Those who desire to acquaint themselves with the steps should 
consult the original papers. We can only present the results, 
and for this purpose have selected the diagram of a specimen 
of Swedish iron, marked No. 101, which is an exact copy of the 
diagram made by the machine. After becoming familiar with 
this one, the student will be able to interpret any of those upon 
the accompanying chart. 

1. The total angle of torsion is marked on the lower horizon- 
tal line. 

2. The ordinates, as bx, represent the moments of torsion, 
and hence represent relatively the strain. 

3. The curve from A to b is convex towards the line of 
abscissas. This shows that the piece had internal strains be- 
fore it was twisted. 

4. From a to b the line is very nearly straight, and the tan- 
gent of its inclination to the horizontal is the ratio of the mo- 
ment of torsion to the total angle of torsion (or distortion) 
which took place between a and b. When the line is exactly 
straight, Hooke's law, " ut tcnsio sic vis" is mathematically 
exact. The inclination of the line is a measure of the stiffness, 
and is proportional to the coefficient of elasticity. The point b, 
where the line begins to curve, corresponds to the limit of elas- 
ticity. When there is no straight part, as in many specimens 
of cast-iron and some other metals, there is not properly any 
limit of elasticity. 

5. The curve, at b, shows that the outer particles begin to 
yield, and a set takes place. 

6. The relative depression at :; shows that the structure of 
the material is not homogeneous. Homogeneous materials 
make a nicely-rounded curve without any depression. This 
simple illustration shows one of the advantages of an automatic 
registry. 



266 



THE RESISTANCE OF MATERIALS. 



|1 

3 £ 
I" 



2* 

o -d 
o a 

i i 



a -a 



o .B 



P. 3 




E1IS 

lie 
■ 

ill 

111 

SI! 






Ilil 



EFFECT OF LONG-CONTINUED STRAINS. 267 

7. The curve rises more rapidly from c to d, showing that 
the previous strains upon the non-homogeneous solid has finally 
brought into action, or at least into greater action, certain ele- 
ments which before were partially dormant. 

8. From d to e the curve is very regular, showing that the 
yielding goes on uniformly. 

9. At e the motion of the lever C was reversed until the arm 
B returned to a vertical position, and the pencil traced the 
curve ef A forward movement was then given to the arms, 
and the pencil traced the curve fg. The abscissa ^/represents 
the set. The inclination of the curve gf as before, represents 
the stiffness at this point. In this case the stiffness is not 
diminished by the previous strains, and the elastic limit has 
been raised. 

10. The strain being continued, the curve gB maintains the 
same height as before, showing that the strength has not been 
impaired ; but at B there is a slight depression. At B the 
arms were fixed in position and the piece was left under strain 
for one day, when the arm C was again moved forward, and 
the line moved suddenly upward to i. 

11. The elevation of the curve Bi was an unexpected dis- 
covery by the experimenter, which was first formally announced 
by him to the American Society of Civil Engineers in Novem- 
ber, 1873. The explanation of this phenomenon is given in his 
own words, as follows : " The phenomenon here observed is in 
elevation of the limit of elasticity by a continued strain. The 
cause is probably a gradual release of internal strain, occurring 
in a somewhat similar manner to that observed previously in 
cast-iron, and less frequently, and generally in a less marked 
degree, in wrought-iron and other metals, which have beeiv 
worked in large pieces, and in which such strain has been 
more or less reduced by a period of rest." 

The piece resisted a much larger strain within the elastic 
limits, after resting twenty-four hours, than it did before ; and 
it evidently required a greater force to rupture it than it would 
if the experiment had continued consecutively on the first day. 
The experimenter, however, concludes that it would probably 
rupture with a less angle of torsion after resting than it would 
have done without resting. We have here a clear proof of the 



268 THE RESISTANCE OF MATERIALS. 

vague inferences which might be drawn from Articles 35, 187, 
and 188. The experiments of Commander Beadsley, which 
were made at a later date, by direct tension, are a striking con- 
firmation of this phenomenon, as given in Article 189. 

12. The curve iC being nearly horizontal shows that the 
resistance remains nearly constant whilst the twisting force is 
active, and the torsional angle is constantly increasing. This 
corresponds to the gradual pulling apart of pliable bodies under 
the action of a nearly constant force, the elongation gradually 
increasing while the section is gradually diminishing, until it 
finally breaks. 

13. At C the strain was maintained constant for another day, 
when we see that another elevation in the elastic limit from 
to k took place. 

14. From k the resistance gradually diminished to D, after 
which it gradually increased to E. At this point a sudden 
movement was given to the lever 6 y , and immediately after the 
movement ceased there was a slight depression in the curve, 
after which the force was gradually applied and the height of the 
curve was gradually restored. The depression in this diagram 
is small compared with those which arose from a similar cause 
in No. 118, as shown on the plate at the points be, Vc' ', and gh. 

This shows that materials will not resist as much to a sudden 
force as they will to one which is applied more gradually. This 
at first appears paradoxical, since it seems to require a larger load 
to break it in a lunger time. But in the light of the preceding 
experiments, and those before referred to, it can be easily ex- 
plained. Suppose, for instance, that 20,000 pounds is just 
sufficient to break a piece if applied all at once ; then if 18,000 
be applied, and after a few hours 500 pounds more be added, 
and after another interval 500 pounds more are added, and so 
on, it will be found that the piece will sustain much more than 
20,000 pounds under the process of slow loading. 

It thus appears that the load which is applied when the action 
is rapid, is not a true measure of the strain ; the more rapid the 
action the less being the strain.* 



* This may be aptly illustrated by a dynamical problem. If a chord passes 
over a pulley, and 100 pounds is attached to each end of it, the tension on the 



EFFECT OF LONG-CONTINUED STRAINS. 209 

This principle lias an important bearing upon the effect oi 
shocks upon machinery. It shows that they are not only weak- 
ened by shocks, as stated in previous articles, but that the mate- 
rial is inherently weaker. It shows that the resilience, or its 
resistance to shock, cannot be correctly determined from the 
work of a statical load in producing rupture. 

After the strain passes the point E, it is not perceptibly di- 
minished until it has been twisted through an angle of 220 
degrees, after which it gradually fails and finally breaks. 

15. The parallelism of the elastic lines shows that the elasti- 
city remains quite unimpaired up to the point of incipient rup- 
ture, a fact which was observed by some of the earliest experi- 
menters in this field of investigation. 

The principles here stated will enable the student to deter- 
mine the qualities of the several specimens shown on the ac- 
companying Plate. 

16. The fact that the resistance remains so nearly uniform 
while the torsional angle is increasing so largely, leads one to 
infer that as the outer elements become weakened by being 
overstrained, that those near the axis resist more, and there 
appears to be a tendency to cause all the elements to resist the 
same amount; so that at the instant of rupture, in ductile 
bodies, the greater part of the transverse section resists uni- 
formly. It is evident that the law given in the Chapter on 
Torsion, that the resistance varies directly as the distance from 
the axis, is not true after a set has taken place. If the resist- 
ance is uniform, we have, in the case of cylinders, nrr^.J for 
the total resistance, in which J is the modulus for ultimate 
shearing, which, in the case of wrought-iron, is nearly the* same 
as T, the tenacity. The mean arm of this force is f r ; hence, 
we have 

Pa = %irr % J. 

Equation (202) is Pa = im*J. 
The former is 1-J times the latter. 

chord will be 100 pounds. But if 50 pounds be removed from one end, accel- 
erated motion will at once take place, and the tension will no longer be 100 
pounds, but it will be 66£ pounds. This is considerably less than the 100 
pounds, the greater weight. The greater the acceleration the less will be the 
tension compared with the load. 



270 THE .RESISTANCE OF MATEEIALS. 

But as it is not probable that the strains can be made uni- 
form, the farmer may be considered the superior limit of the 
strength. 

17. The diagrams on the upper half of the Plate represent 
the strains on several specimens at various temperatures. The 
general conclusion arrived at by these experiments was, that 
with pure, well-worked metals, a diminution of temperature 
produces an increase of strength, but when there is an excess 
of impurities this law may be reversed, especially in case of 
shocks. 



LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 271 



CHAPTER XL 

LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 

194. risk and safety. — We have now considered the 
breaking-strength of materials under a variety of conditions, 
and also the changes produced upon them when the strains are 
within the elastic limits. In a mechanical structure, in which 
a single piece, or a combination of pieces, are required to sus- 
tain a load, it is desirable to know how small the piece, or the 
several pieces, may be made to sustain a given load safely for 
an indefinite time ; or, how much a given combination will sus- 
tain safely. The nature of the problem is such that an exact 
limit cannot be fixed. Materials which closely resemble each 
other do not possess exactly the same strength or stiffness ; and 
the conditions of the loading as to the amount or manner in 
which it is to be applied, may not be exactly complied with. 
Exactness, then, is not to be sought ; but it is necessary to find 
a limit below which, in reference to the structure, or above 
which, in reference to the load, it is not safe to pass. 

It is evident that to secure an economical use of the material 
on the one hand, and ample security against failure on the 
other, the limit should be as definitely determined as the nature 
of the problem will admit ; but in any case we should incline 
to the side of safety. No doubt should be left as to the stabil- 
ity of the structure. There is no economy in risk in perma- 
nent structures. Risk should be taken only in temporary, or 
experimental, structures ; or where risk cannot, from the nature 
of the case, be avoided. 

19o. absolute MODULUS of safety. — In former times, 
one of the principal elements which was used for securing 
safety in a structure, was to assume some arbitrary value for 
the resistance of the material, such value being so small that 



272 THE JIERIRTANCE OF MATERIALS. 

the material could, in the opinion of the engineer, safely sus- 
tain it. Tins is a convenient mode, but very unphilosophical, 
although still extensively used. The plan was to determine, as 
nearly as possible, what good materials would sustain For a long 
period, and use that value for all similar materials. But it is 
evident, from what has been said in the preceding pages, that 
some materials will sustain a much larger load than the aver- 
age, while others will not suslain nearly so much as the aver- 
age. In all sueh cases the proper value of the modulus can 
only be determined by direct experiment. In all important 
structures the strength of the material, especially iron and steel, 
should he determined by direct experiment. 

The following values are generally assumed for the modulus 
of safety. 

Pounds yior square inch. 

Wrong-ht-iron, for tension or compression, from 10,000 to 12,000 

Cast-iron, for tension, from 3,000 to 4,000 

Cast-iron, for compression, from 15,000 to 20,000 

Wood, tension or compression, from 850 to 1,200 

i granite, from 400 to 1,200 
quartz, from 1,200 to 2,000 
sandstone, from 300 to 600 
limestone, from 800 to 1,200 

The practice of French engineers,* in the construction of 
bridges, is to allow 3.8 tons (gross) per inch upon the gross sec- 
tion, both for tension and compression of wrought iron. 

The Commissioners on Railroad Structures, England, estab- 
lished the rule that the maximum tensile strain upon any part 
of a wrought iron bridge should not exceed live tons (gross) 
per square inch.f 

In most eases the effective section is the section which is sub- 
jected to the strain considered. 

190. factor of safety. — The next mode, and one which 
is also largely in use, is to take a fractional part of the ultimate 
strength of the material for the limit of safety. The recipro- 
cal of this fraction is called th e factor of 'safety. It is the ratio 
of the ultimate strength to the computed strain, and hence is 

* Am. R. R. Times, 1871, p. 6. 

\ Civ. Eng. and Arch. Jour., Vol. xxiv., p. 327 



LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 273 

(lie favtor by which the computed strain must be multiplied to 
equal the actual strength of the material, or of the structure. 

Experiments and theory combine to teach that the factor of 
safety should not be taken as small as 2. See articles 19, 1S5, 
l<Sf,, and 188. 

Beyond this the factor is somewhat arbitrarily assumed, de- 
pending upon the ideas of the engineer. For instance, the fol- 
lowing values were given to the Commissioners on Kail way 
Structures in England.* 

Factors. 

Messrs. May and Grissel 3 

Mr. Brunei'. 3 to 5 

Messrs. Rasbrick, Barlow and others 6 

Mr. 1 1 awkshaw 7 

Mr. Glyn 10 

The following values are also given by others : — 

Factors. 

Bow, for wrought-iron beams 3.5 

Weisbach, for wrought-iron f 3 to 4 

Vicat, for wire suspension bridges more than 4 

r> i • £ • i -j j steady strain 3 to 4 

Kan km e, ror wire bridges < . -, -, 

& I moving load 6 to 8 

tt,. , . , . , ( for posts and braces 5 to 6 

1^ ink, iron-truss bridges. . \ _ r . . . ^ 

( for cast-iron chords 7 

Fairbairn, for cast-iron beams X 5 to 6 

C. Shaler Smith, compression of cast-iron 5 

Rankine and others, Cor cast-irou beams 4 to 6 

Mr. Clarke, in Quincy Bridge, lower chord 6 to 7 

Washington A. Roebling, for suspension cables 6 

Morin, Yicat, Weisbach, Rondelet, Kavier, Barlow, and 

many others, say that for a wooden frame it should 

not be less than 10 

For stone, for compression 10 to 15 

From the experiments which are recorded in Article 188, 
Fairbairn deduced the following conclusions in regard to beams 

* Civ. Eng. and Arch. Jour. , Vol. xxiv. , p. 327. 
\ Weisbach, Mech. and Eng., Vol. i., p. 201. 
% Fairbairn, Cast and Wrought-iron, p. 58. 



274 THE RESISTANCE OF MATERIALS. 

and girders, whether plain or tubular.* " The weight of the 
girder and its platforn should not in any case exceed one-fourth 
the breaking weight, and that only one-sixth of the remaining 
three-fourths of the strength should be used by the moving 
load." According to this statement the maximum load, includ- 
ing the live and dead load, may equal, but should not exceed, 

of the breaking load. Hence the factor of safety must not be 
less than 2.66 when the above conditions are fulfilled. This 
value is, however, evidently smaller than is thought advisable 
by most engineers. 

The rule adopted by the Board of Trade, England, for rail- 
road bridges is f " to estimate the strain produced by the great- 
est weight which can possibly come upon a bridge throughout 
every part of the structure which should not exceed one-fifth 
the ultimate strength of the metal" They also observed that 
ordinary road bridges should be proportionately stronger than 
ordinary railroad bridges. 

197. rational limit of safety. — It is evident that 
materials may. be strained any amount within the elastic limit. 
Their recuperative power — if such a term may properly be 
used in connection with materials — lies in their elasticity. If 
that is damaged the life of the material is damaged, and its 
powers of resistance are weakened. As we have seen in the 
preceding pages, there is no known relation between the coeffi- 
cient of elasticity and the ultimate strength of materials. The 
coefficient of elasticity may be high and the modulus of strength 
comparatively low. In other words, the limit of elasticity of 
some metals may be passed by a strain of less than one-third 
their ultimate strength, while in others it may exceed one-half 
their ultimate strength. We see, then, the unphilosophical 
mode of fixing an arbitrary modulus of safety, or even a fac- 
tor of safety, when they are made in reference to the ultimate 
strength. But an examination of the results of experiments 

* Civ. Eng. and Arch. Jour., Vol. xxiv., p. 329. 
\ Civ. Eng. and Arch. Jour., Vol. xxiv., p. 226. 



LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 275 

shows that the limit of elasticity is rarely passed for strains 
which are less than one-third of the ultimate strength of the 
metal, and hence, according to the views of the engineers given 
in the preceding article, the factors of safety, which are com- 
monly used in practice, are generally safe. But if the limit of 
elasticity were definitely known it is quite possible that a 
smaller factor of safety might sometimes be used.* 

This method of determining the limit has been recognized by 
some writers, and the propriety of it has been admitted by many 
practical men, but the difficulty of determining the elastic limit 
has generally precluded its use. The experiments which are 
necessary for determining it are necessarily more delicate than 
those for determining the ultimate strength. 

In regard to the margin that should be left for safety, much 
depends upon the character of the loading. If the load is 
simply a dead weight, the margin may be comparatively small ; 
but if the structure is to be subjected to percussive forces or 
shocks, it is evident, as indicated in articles 19 and 193, that 
the margin should be comparatively large, on account of the 
indeterminate effect produced by the force. In the case of 
railroad bridges, for instance, the vertical posts or ties, as the 
case may be, are generally subjected to more sudden strains due 
to a passing load, than the upper and lower chords, and hence 
should be relatively stronger. The same remark applies to the 
inclined ties and braces which form the trussing ; and to any 
parts which are subjected to severe local strains. 

In machines which are subjected to a constant jar while in 
use, it is very difficult to determine the proper margin which 
is consistent with economy and safety. Indeed, in such cases, 
economy as well as safety generally consists in making them 
excessively strong, as a single breakage may cost much more 
than the extra material necessary to fully insure safety. 

The mechanical execution of a structure should be taken 
into consideration in determining the proper value of the mar- 
gin of safety. If the joints are imperfectly made, excessive 

* James B. Eads, in his Report upon the Illinois and St. Louis Bridge, for 
.871, states that he tested samples of steel which were to be used in that 
structure, which showed limits of elastic reaction of 70,000 to 93,000 pounds 
per square inch. 



276 THE RESISTANCE OF MATERIALS. 

strains may fall upon certain points, and to insure safety the 
' margin should be larger. ISTo workmanship is perfect, but the 
elasticity of materials is favorable to such imperfections as 
necessarily exist ; for, when only a portion of the surface which 
is intended to resist a strain is brought into action, that por- 
tion is extended or compressed, as the case may be, and thus 
brings into action a still larger surface. 

198. EXAMPLES OF STRAINS THAT HAVE BEEN USED 

in practical cases. — The margin of safety that has been 
used in various structures may or may not serve as guides 
iu designing new structures. If the margin for safety is so 
small that the structure appears to be insecure and gives indi- 
cations of failure, it evidently should not be followed. It 
serves as a warning rather than as a guide. If the margin 
is evidently excessively large, demanding several times the 
amount of material that is necessary for stability, it is not a 
guide. Any engineer or mechanic, without regard to scientific 
skill or economy in the use of materials, may err in this direc- 
tion to any extent. But if the margin appears reasonably safe, 
and the structure has remained stable for a long time, it serves 
as a valuable guide, and one which may safely be followed 
under similar circumstances. Structures of this kind are 
practical cases of the approximate values of the inferior limits 
of the factors of safety. The following are some practical 
examples : — 



LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 277 



IRON TRUSSED BRIDGES. 



NAME OF THE BRIDGE. 



Passaic {Lattice) 

Place de l'Europe {Lattice) 

Canastota {N. Y. G. R. R.) {Lattice). . 

Newark Dyke ( Warren Girder) 

Boyne Viaduct {Lattice) 

Charing Cross {Lattice) 

St. Charles, Mo. ( Whipple Truss)* 

Louisville, Ky. ( Fink Truss) 

Keokuk and Hannibal f 

Quincy Bridge % 

Kansas City Bridge § 

Hannibal Bridge ( Quadrangular Truss) 



Tons 
square inch. 



5ito6 

4 

5 
5 
5 
5 

Pounds 

per square: inch. 

12,000 

7,000 to 12,000 

9,251 

10,000 

11,375 

Factor of safety, 5 



Tons 
per square inch. 



4ito5i 
3f 
4 
5 



Pounds 
per square inch. 

12,000 
\ to \ the strength 

8,962 
Factor of safety, 5 

711 
Factor of safety, 5 



WOODEN BRTDGES. 



NAME OP THE BRIDGE. 


MAXIMUM STRAINS. 


Cumberland Valley B,. B. Bridge 


635 pounds per square inch. 
Factor of safety, 20. 


Portage Bridge {N. T. & E. R. R) 





* R. R. Gazette, July 8, 1871, p. 169. 

f R. R. Gazette, July 15, 1871, p. 178. Pivot span 376 feet 5 inches : longest 
pivot span yet constructed. 

% Report of Chief Engineer Clark. 

§ Calculated from the Report of Chief Engineer O. Chanute, pp. 106 and 
136. 

| The tensile strength of the material ranged from 55,000 lbs. to 65,000 lbs, 
per square inch. — R. R. Gazette, July 15, 1871, p. 169. 



278 



THE RESISTANCE OF MATERIALS. 



CAST-IRON ARCHES.* 



NAME OP THE AKCH. 


SPAN. 

Feet. Inches. 


VERSED SINE. 

Feet Inches. 


STBAIN 

PEE SQUARE 

INCH IN TONS. 


Austerlitz 


186 
152 2 
102 5 
137 9 
197 10 
120 


10 7 
16 1 

11 4 

15 

16 5 
20 


2.78 
1.46 
1.37 
1.90 
2.37 
3.00 


Carrousal 


St. Denis 


Nevers 


Rhone 


Westminster 





STONE ARCHES, f 



NAME OP THE AKCH. 


Span in feet. 


Versed sine 
in feet. 


Pressure per 

square inch in 

pounds at the 

key. 


Factor of safety 

at the point of 

greatest 

strain. 


Wellington 


100 

120 
128 
140 
147 
152 
200 


15 
35 
32 

35 

18 
38 
42 


175 
151 

172 
244 
293 
215 
349 


11.3 


Waterloo (9 Arches) . . . 
Neuilly 


20.0 
11.6 


Taaf (Smth Wales)... 
Turin 


8.0 
10.2 


London 


14 


Chester 


8 6 







CAST-STEEL ARCH. 



NAME OF THE AECH. 


SPAN. 
Feet. 


FACTOR 
OF SAFETY. 


Illinois and St. Louis Bridge 


515 


6 + * 





* Irwin on Iron Bridges and Roofs. 

f Cresy's Encyclopedia. 

\ Report of the Engineer, p. 33. 



LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 279 



SUSPENSION BRIDGES. 



NAME OF THE BBLDGE. 


.5 

a 
a 
m 


-Sga 
IN 


fl§o 


I 

o 

t 

[=4 


Menai 


580 

422i 

666 

384 

702£ 

821 

160 

1,057 
1,600§ 

1,665 fl 


4.21 
5.38 
5.01 
4.36 
2.90 
6.70 

9.1 


8.00 
9.36 
8.11 
8.07 
5.03 
8.40 

11.7 


3 9* 


Hammersmith 


3.3* 


Pesth 


3 9* 


Chelsea 


3 9* 


Clifton 


6 4* 


Niagara 


5.3f 

4 
6.2 


Suspension Aqueduct, Pitts- ) 

burgh. Pa. 7 spans each, j 

Cincinnati Bridge f 


East River 


6.0 


Highland (proposed) 


6.0 







TUBULAR BRIDGES. 





SPAN. 

Feet. 


FOR WEIGHT OF BRIDGE 
AND LOAD. 


NAME OF THE BEIDGE. 


Tension. 
Tons. 


Compression, 
Tons. 


Conway 


400 
460 


6.85 

3.00 
4.75 


5 03 


Britannia (Central span) 




Penrith (Tubular Girder) 


4 25 







* Tensile strength, 70,000 lbs. per square inch. 

f Tensile strength, 100,000 lbs. per square inch. 

X Report of the Chief Engineer, J. A. Roebling. 

§ Engineer's Report. Suspending ties, factor of safety, 8. 

| Jour. Frank. Inst, voL lxxxyii., p. 165. 



280 



THE RESISTANCE OF MATERIALS. 
STONE FOUNDATIONS. 





FACTOE OF SAFETY. 


Pillars of the Dome of St. Peter's {Borne) 


16 


" " St. Paul's {London) 


14 


" " St. Genevieve {Paris) 


7.6 


Pillars of the Church Toussaint {Angers)* 


10 


Merchants' Shot Tower {Baltimore) 


4.8 


Lower courses of Britannia Bridge 


31 


Lower courses of the Piers of Neuilly Bridge {Paris) 

Foundation of St. Charles' Bridge {Missouri) 
Foundations of East River Bridge f 


15.8 
12 to 14 
10 to 20 







199. proof load. — The proof load is a trial load. It 
is intended as a practical test of a structure. 

It generally exceeds the greatest load that it is ever intended 
to put upon the structure when in actual service. 

According to the principles which have been discussed in 
the preceding pages, it is evidently better for the structure, and 
should be more satisfactory, to apply a moderate proof load 
for a long time than an excessive one for a short time. 

* Strength of Materials, J. K. Whtidin, p. 23. 

f " In the stone work the pressures vary from 8 to 26 tons per square foot. 
Stone used is granite, selected samples of which have borne a crushing strain 
of 600 tons per square foot. Some will not bear over 100 tons per square foot. 
The general average is necessarily much less than that of the best specimens." 
— Statement of the Chief Engineer \ Washington A. Roebling. 



APPENDIX I. 



PRESERVATION OF TIMBER. 

1. CONDITIONS. — Timber may be subjected to the following condi- 
tions : — 

It may be kept constantly dry ; at least, practically. 

It may be constantly wet in fresh water. 

It may be constantly damp. 

It may be alternately wet and dry. 

It may be constantly wet in sea water. 

2. TI31BER kept CONSTANTLY dry will last for centuries. The roof 
of Westminster, Hall is more than 450 years old. In Stirling Castle are carv- 
ings in oak, well preserved, over 300 years old ; and the trusses of the roof of 
the Basilica of St. Paul, Rome, were sound and good after 1,000 years of 
service. * The timber dome of St. Mark, at Venice, was in good condition 
850 years after it was built, f 

Artificial preservatives seem to be unnecessary under this condition. 

3. TIMBER KEPT CONSTANTLY WET IN FRESH WATER, under 
such conditions as to exclude the air, is also very durable. The pillars upon 
which dwellings of the Canaries rest were put up in their present place in 1402, 
and they remain sound to the present time.:}: The utensils of the lake dwell- 
ings of Switzerland are supposed to be at least 2,000 years old.§ 

The piles of the old London Bridge were sound 800 years after they were 
driven. The piles of a bridge built by Trajan, after having been driven more 
than 1,600 years, were found to be petrified four inches, the rest of the wood 
being in its ordinary condition. § 

Beneath the foundation of Savoy Palace, London, oak, elm, beech, and 
chestnut piles and planks were found in a perfect state of preservation after 
having been there 650 years. § 

While removing the old walls of Tunbridge Castle, Kent, there was found 

* The London Builder, vol. ii., p. 616. 
t Modern Carpentry, Silloway, p. 40. 
% Journal of the Frank. Inst., 1870. 
§ Modern Carpentry, Silloway, p. 39. 



282 APPENDIX I. 

in the middle of a thick stone wall, a timber cnrb which had been enclosed 
for 700 years.* 

It is doubtful if artificial preparations would have prolonged the life of the 
timber 1 in these cases. 

4. timber iiv DAMP situations. — Timber, in its native state, under 
these circumstances, is liable to decay rapidly from the disease called " dry 
rot." In dry rot the germs of the fungi are easily carried in all directions in 
a structure where it has made its appearance, without actual contact between 
the sound and decayed wood being necessary ; whereas the communication of 
the disease resulting from wet rot takes place only by actual contact. The 
fungus is not the cause of the decay, but only converts corrupt matter into 
new forms of life, f 

There are three conditions which are at our command for prolonging the 
life of timber in damp situations : — 

1st. Thoroughly season it ; 

2d. Keep a constant circulation of air about it ; and 

3d. Cover it with paint, varnish or pitch. 

The first condition is essential, and may be combined with either or both 
of the others. 

The cellulose matter of the woody fibre is very durable when not acted upon 
by fermentation, and it is this that we desire especially to protect. The water 
of the sap and the albuminous substances in timber are fermentable sub- 
stances, and when both are present they are ever ready, under suitable circum- 
stances, to promote decay. By seasoning we do not mean simply drying so 
as expel the water of the sap, but also a removal or change of the albumi- 
nous substances. 

Unseasoned timber which is surrounded by a dead air decays very rapidly. 
The timber of many modern constructions is translated from the forests and 
enclosed in a finished building in a few weeks, and unless it is subject to a free 
circulation of air it inevitably decays rapidly. :j: 

Thorough ventilation is indispenable to the preservation of even well-sea- 
soned naked wood in damp localities. The rapid decomposition of sills, sleep- 
ers, and lower floors is not surprising where neither wall-gratings nor venti- 
lating flues carry off the moisture rising from the earth, or foul gases evolved 
in the decay of the surface mould. In the close air of cellars, and beneath 
buildings, the experiments of Pasteur detected the largest percentage of 
fungi spores. If the earth be removed to the foot of the foundation, and the 

* Modern Carpentry, Silloway, p. 39. 

t " There is no reason to believe that fungi can make use of organic compounds in any other 
than a state of decomposition." — Carpenter's Oomp. Physiology, p. 165. (See also Encyclopaedia 
Britanica on this subject. 

X For an account of the rapid destruction of the floors and joists of the Church of the Holy 
Trinity, Cork, Ireland, by dry rot, see Civil Engineer's Journal, vol. xii., p. 303. For an ac- 
count of the decay of floors, studs, &c.. in a dwelling, see the London Builder, vol. vi., p. 34. 

"In some of the mines in France the props seldom last more than fifteen months." — Annales 
des Mines. 



APPENDIX I. 283 

cavity filled with dry sand, plaster-rubbish, etc. , or if a thick stratum of cement 
be laid down to exclude the water, and a complete circulation of air be pro- 
vided for, the lower floors will last nearly as long as upper ones. * 

A covering of paint, pitch, varnish, or other impervious substance upon un- 
dried timber is very detrimental, for by it all the elements of decay are re- 
tained and compelled to do their destroying work. The folly of oiling, paint- 
ing, or charring the surface of unseasoned wood is therefore evident. Owing 
to this blunder alone, it is no unusual thing to find the painted wood- work of 
older buildings completely rotted away, while the contiguous naked parts are 
perfectly sound. 

While an external application of coal tar promotes the preservation of dry 
timber, nothing can more rapidly hasten decay than such a coating upon the 
surface of green wood. But this mistake is often made, and dry rot does the 
work of destruction.! Carbonizing the surface also increases the durability of 
dry, but promotes the decay of wet timber. Farmers very often resort to 
one of the latter methods for the preservation of their fence-posts. Unless 
they discriminate between green and seasoned timber, these operations will 
prove injurious instead of beneficial. 

There are numerous methods for promoting the process of seasoning. 
Some have in view simply drying, a process which is important in itself, but 
which will not in itself prevent decay in damp situations unless the moisture 
be permanently excluded. Some dry with hot air, and some with steam. In 
the latter case, if the steam be superheated the process is very rapid, but it 
seems to damage the life of the timber. 

Others have in view the expulsion of the albuminous substance. Water-soak- 
ing the logs and afterwards drying the lumber, seems to be a cheap and quite 
effectual mode. But there are many patented processes for securing this end, 
or for changing the albuminous substances ; and in many cases the latter end 
is not only secured, but the salts which are used act directly upon the cellulose 
and lignite of the wood, thereby greatly promoting its durability. 

The following are the principal processes which have been used : — Mr. P. 
W. Barlow's patent \ provided for exhausting the air from one end of the log, 
While one or more atmospheres press upon the other end. This artificial 
aerial circulation through the wood is prolonged at pleasure. However excel- 
lent in theory, this process is not practicable. 

By another method, the smoke and hot gases of a coal fire are conveyed 
among the lumber, placed in a strong draft. Some writers recommend the re- 
moval of the bark one season before felling the tree. All good authorities 
agree that the cutting should take place in the winter season. § 

* The Builder, vol. xi., page 46. 

t According to Col. Berrien, the Michigan Central Railroad bridge, at Niles, was painted, 
before seasoning, with " Ohio fireproof paint,''' forming a glazed surface. About five years after 
it was so badly dry-rotted as to require rebuilding. 

% Civ. Eng. Jour., vol. xix., p. 422. 

§ Experiments detailed in the Cosmos show conclusively that winter-cut pine is stronger and 
more durable than that cut at any other season of the year. — Ann. Sc. Discovery for 1861, p. 346 

" Oak trees felled in the winter make the best timber." — The Builder, 1859, page 138. 



284 APPENDIX I. 

Kyan's process, which consists in the use of corrosive sublimate, was pat 
ented in 1832. His specific solution * was one pound of chloride of mercury 
to four gallons of water. Long immersion in the liquid in open vats, or great 
pressure upon both solution and wood, in large wrought-iron tanks, is neces- 
sary for the complete injection of the liquid. The durability of well kyanized 
timber has been proved, but the expensiveness of the operation will long for- 
bid its extensive adoption. 

For '* Burnettizing," f a solution of chloride of zinc — one pound of salt to 
ten gallons of water — is forced into the wood under a pressure of 150 lbs. per 
square inch. 

Boucherie employs a solution % of sulphate of copper, one pound of the sul- 
phate to twelve and a half gallons of water, or one gallon of pyrolignite of iron 
to six gallons of water. He enclosed one end of the green stick in a close-fit- 
ting collar, to which is attached an impervious bag communicating through a 
flexible tube with an elevated reservoir containing the salt liquid. Hydrostatic 
pressure soon expelled the sap at the opposite end of the log; and when the 
solution also makes its appearance the process is completed. 

He found that the fluid will pass along the grain, a distance of 12 feet, un- 
der a lower pressure than is required to force it across the grain, three-fourths 
of an inch. The operation is performed upon green timber with the greatest 
facility. § 

In 1846, eighty thousand sleepers of the most perishable woods, impregnated, 
by Boucherie's process, with sulphate of copper, were laid down on French 
railways. After nine years' exposure, they were found as perfect as when laid. |j 
This experiment was so satisfactory that most of the railways of that empire 
at once adopted the system. We would suggest washing out the sap with 
water, which would not coagulate its albumen. The solution would appropri- 
ately follow. 

Both of the last-named processes are comparatively cheap, ^f 



* Civ. Eng. Jour., vol. v., page 202. 

+ Civ. Eng. Jour., vol. xiv., p. 471. Invented by Burnett in 1838. 

X Civ. Eng. Jour., vol. xx., p. 405. 

§ As a modification of this method he also cut a channel in the wood throughout the circumfer- 
ence of the tree, fitted a reservoir thereunto, and poured in the liquid. The vital forces speedily 
disseminated the solution throughout the tree. 

|| Jour, of the Frank. Inst., vol. xxxii., pp. 2, 3. 

^The manufacturing companies of Lowell, Massachusetts, have an establishment for " Bur 
nettizing " timber, in which they prepare sticks fifty feet in length. Under a pressure of 125 
pounds per square inch they inject from two to eight ounces of the salt into each cubic foot of 
wood. The cost, in 1861, was from $5 to $> 6 per 1,000 feet, board measure. 1 Boucherie's method 
must be still cheaper. It costs less than creosoting by one shilling per sleeper. 2 — New Am. Cycl. 

An American engineer, Mr. Hewson, for injecting railroad sleepers, proposes a vat deep 



1 The Philadelphia, Wilmington and Baltimore Railroad Company have used the process since 
1860 with complete success. The Union Pacific Railroad Company have recently erected a large 
building for this purpose. Their cylinder is 75 feet long, 61 inches in diameter, and capable of 
holding 250 ties. They "Burnettize " two batches per day.— Report on Pacific Railroad, by CoL 
Simpson, 1865. 

3 Jour. Frank.. Inst., vol. xxxii., pp. 2, 3. 



APPENDIX I. 285 

By Earie's process the timber is boiled in a solution of one part of sulphate 
of copper, three parts of the sulphate of iron, and one gallon of water to 
every pound of the salts. A hole was bored the whole length of the piece 
before it was boiled. It was boiled from two to four hours, and allowed to 
cool in the mixture. 

Ringold and Earle invented the following process : — A hole was made the 
whole length of the piece, from one-half to two inches in diameter, and 
boiled from two to four hours in lime-water. After the piece was dried the 
hole was filled with lime and coal tar. Neither of these methods was very 
successful. 

A Mr. Darwin suggests that the piece be soaked in lime-water, and after- 
wards in sulphuric acid, so as to form gypsum in the pores. 

Bethell's process consists in forcing dead oil into the timber. This is 
called creosoting.* He inclosed the timber and dead oil in huge iron tanks, 
and subjected them to a pressure varying between 100 and 200 pounds per 
square inch, at a temperature of 120° F. about twelve hours. From eight to 
twelve pounds of oil are thus injected into each cubic foot of wood. Lumber 
thus prepared is not affected by exposure to air and water, and requires no 
painting, f A large number of English railway companies have already 
adopted the system. % Eight pounds of oil per cubic foot is sufficient for 
railway sleepers. § 

One writer has said that if creosote has ever failed to prevent decay, it has 
been because of an improper treatment, or because the oil was deficient in 
carbolic acid. 

Sir Robert Smirke was one of the first architects to use this process, and 
when examined before a Committee on Timber, stated that this process does 
not diminish the strength of the material which is operated upon. He after- 
wards said, " I cannot rot creosoted timber, and I have put it to the severest 
test I could apply." 

The odor of creosote makes it objectionable for residences and public 
buildings. 

Mr. S. Beer, of New York City, invented a mode of preserving timber by 



enough for the timbers to stand in upright. The pressure of the surrounding solution upon the 
lower ends of the sticks will, he thinks, force the air out at their upper extremities, kept just 
above the surface of the solution, after which the latter will rise and impregnate the wood. In 
1859 he estimated chloride of zinc at 9 cents per pound, sulphate of copper at 14 cents per pound, 
and pyrolignite of iron at 23 cents per gallon. He found the cost of impregnating a railway tie 
with sufficient of those salts to prevent decay, to be : for the chloride of zinc 2-8 cents, for blue 
vitriol 3*24 cents, for pyrolignite of iron 7*5 cents. — Jour. Frank. Inst., vol. xxvii., p. 8. 

* Creosote from coal undoubtedly contains two homologous bodies, C 12 H 6 2 and C 14 H 8 2 , 
the first being carbolic and the second cresylic acid."*— Ore's Diet, of Arts, Manu., and Mines, vol. 
ii., p. 623. 

t "Ore's Diet, of Manu. and Mines. 

X The Great Western, North-Eastern, Bristol and Exeter, Stockton and Darlington, Manchester 
and Birmingham, and London and Birmingham. — Ore's Diet, of Manu, and Mines. 

§ Jour. Frank. Inst., vol. xliv., p. 275. 



286 APPENDIX I. 

boiling it in borax with water. But this process has been objected to on the 
ground that it is not a good protection against moisture. 

Common salt is known to be a good preservative in many cases. According 
to Mr. Bates's opinion, * it answers a good purpose in many cases if the pieces 
to which it is exposed are not too large. 

5. TIMBER ALTERNATELY WET AND DRY. — The Surface of all 

timber exposed to alternations of wetness and dryness, gradually wastes away, 
becoming dark-colored or black. This is really a slow combustion, but is com- 
monly called wet rot, or simply rot. Other conditions being the same, the 
most dense and resinous woods longest resist decomposition. Hence the su- 
perior durability of the heart-wood, in which the pores have been partly filled 
with lignine, over the open sap-wood, and of dense oak and lignum-vitae 
over light poplar and willow. Hence, too, the longer preservation of the 
pitch-pine and resinous " jarrah " of the East, as compared with non-resinous 
beech and ash. 

Density and resinousness exclude water. Therefore our preservatives 
should increase those qualities in the timber. Fixed oils fill up the pores and 
increase the density. Staves from oil-barrels and timbers from whaling ships 
are very durable. The essential oils resinify, and furnish an impermeable coat- 
ing. But pitch or dead oil possesses advantages over all known substances for 
the protection of wood against changes of humidity. According to Professor 
Letheby,f dead oil, 1st, coagulates albuminous substances; 2d, absorbs and 
appropriates the oxygen in the pores, and so protects from eremacausis ; 3d, 
resinifies in the pores of the wood, and thus shuts out both air and moisture ; 
and 4th, acts as a poison to lower forms of animal and vegetable life, and so pro- 
tects the wood from all parasites. All these properties specially fit it for im- 
pregnating timber exposed to alternations of wet and dry states, as, indeed, 
some of them do, for situations damp and situations constantly wet. Dead oil 
is distilled from coal-tar, of which it contains about .30, and boils between 
390° and 470° Fahr. Its antiseptic quality resides in the creosote it contains. 
One of the components of the latter, carbolic acid (phenic acid, phenol), 
Ci 2 H 6 2 , the most powerful antiseptic known, is able at once to arrest the de- 
cay of every kind of organic matter:^ Prof. Letheby estimates this acid at £ 

* Report of the Commissioner of Agriculture. 

t Civ. Eng. Jour., vol. xxiii., p. 216. 

X " I have ascertained that adding one part of the carbolic acid to five thousand parts of a 
strong solution of glue will keep it perfectly sweet for at least two years. . . . . . . Hidea 

and skins, immersed in a solution of one part of carbolic acid to fifty parts of water, for twenty- 
four hours, dry in air and remain quite sweet." — Prof. Crace Calvert, Ann. Sc. Discov., 1865, 
p. 55. 

" Carbolic acid is sufficiently soluble in water for the solution to possess the power of arresting 
or preventing spontaneous fermentation. Saturated solutions act on animals and plants as a 
■virulent poison, though containing only five per cent, of the acid." — Civ. Eng. Jour., vol. xxii., 
p. 216. 

" Parasites and other worms are instantly killed by a solution containing only one-half per cent. 
of acid, or by exposure to the air containing a small portion of the acid Bj 



APPENDIX I. 



287 



to 6 per cent, of the oil. Cresylic acid Ci 4 H 6 2 , the homologue of carbolic 
acid, and the other component of creosote, is not known to possess preserva- 
tive properties. 

Creosoting, or Beth ell's process, is the most valuable of all the well-tried 
processes in this case. For railway sleepers eight pounds of oil per cubio 
foot of timber is sufficient. * If the timber is dry, a coating of coal-tar, paint, 
or resinous substance, is valuable. 

A Mr. Heinrnann, of New York City, proposes the following process, which 
appears to be very promising : — 

The sap is first expelled and then the timber is injected with common rosin. 
The latter is introduced while in a liquid state, under high pressure, while in 
vessels especially constructed for the purpose. 

In an experiment made by Prof. Ogden, one cubic foot of green wood ab- 
sorbed 8.96 pounds of rosin, while a cubic foot of well-seasoned wood absorbed 
only 2.66 pounds. The strength of the timber was increased by this process, 
as is shown by the following experiment : — 



Wood treated with Rosin. 


Wood in its Natural State. 


Breaking 

Weight. 

Pounds. 


Quality. 


Grain. 


Breaking 

Weight. 

Pounds. 


Quality. 


Grain. 


163.5 

193.0 

171.5 

72.5 

57.5 

57.5 

121.0 

155.5 


Checked. 

Sound, 
it 

Checked. 
<< 

a 
u 
u 


Straight. 
n 

u 

Cross. 
Slant. 

k 
Cross. 


98.5 

103.0 

116.0 

57.5 

46.00 

46.0 

71.0 

84.0 


Sound. 


Slant. 
Straight. 



It is found by experiment that wood thus treated is not as flammable as air- 
dried wood. This is accounted for from the fact that a kind of inflammable 
slag is deposited over the surface immediately after the rosin begins to burn. 

The chief advantages which are claimed for this method are more theoreti- 
cal than practical, as it has not yet had sufficient time to test its practical 
merits, and it may, like many other processes, disappoint the hopes of its 
strongest advocates and well-wishers. 



examining the action on a leaf, we find the albumen is coagulated. All animals with a naked skin, 
and those that live in water, die sooner than those that live in air and have a solid envelope."— 
Dr. I. Lemaire, Ann. Sc. Discov., 1855, p. 23S. 
* Jour. Frank. Inst. vol. xliv., p. 275. 



APPENDIX I. 

6. TIMBER COWSTANTIiY WET US' SALT WATER.-We have not to 
guard against decay when timber is in this situation. Teredo navalis, a mol- 
lusk of the family Tubicolaria, Lam., soon reduces to ruins any unprotected 
submarine construction of common woods. We quote from a paper read before 
the "Institute of Civil Engineers," England, illustrating the ravages of this 
animal: — 

" The sheeting at Southend pier extended from the mud to eight feet above 
low-water mark. The worm destroyed the timber from two feet below the 
surface of the mud to eight feet above low-water mark, spring-tide ; and out 
of 38 fir-timber piles and various oak-timber piles, not one remained perfect 
after being up only three years." * Specimens of wood, taken from a vessel 
that had made a voyage to Africa, are in the museum, and show how this rapid 
destruction is effected. 

i ' None of our native timbers are exempt from these inroads. Robert Ste- 
phenson, at Bell Rock, between 1814 and 1843, f found that green heart oak, 
beef -wood, and bullet-tree were not perforated, and teak but slightly so. 
Later experiments show that the " jarrah " of the East, also, is not attacked.^: 
The cost of these woods obliges us to resort to artificial protection. 

" The teredo never perforates below the surface of the sea-bottom, and prob- 
ably does little injury above low-water mark. Its minute orifice, bored across 
the grain of the timber, enlarges inwards to the size of the finger, and soon be- 
comes parallel to the fibre. The smooth circular perforation is lined through- 
out with a thin shell, which is sometimes the only material separating the 
adjacent cells. The borings undoubtedly constitute the animal's food, por- 
tions of woody fibre having been found in its body.§ While upon the surface 
only the projecting siphuncles indicate the presence of the teredo, the wood 
within may be absolutely honey-combed with tubes from one to four inches 
in length. 

" It was naturally supposed that poisoning the timber would poison or drive 
away the teredo, but Kyan's, and all other processes employing solutions of 
the salts of metals of alkaline earths signally failed. This, however, is not 
surprising. The constant motion of sea- water soon dilutes and washes away 
the small quantity of soluble poison with which the wood has been injected . 
If any albuminate of a metallic base still remains in the wood, the poisonous 
properties of the injection have been destroyed by the combination. More- 
over, the lower vertebrates are unaffected by poisons which kill the mammals. 
Indeed, it is now known that certain of the lower forms of animal life live and 
even fatten on such deadly agents as arsenic. | 

" Coatings of paint or pitch are too rapidly worn away by marine action to 



* Civ. Eng. Jour. vol. xii., p. 382. 

t The Builder for 1862, p. 511. 

i Civ. Eng. Jour., vol. xx„ p. 17. 

§ Civ. Eng. Jour., vol. xii., p. 382. Also Diet. Univ. tfHist. Natur. tome xii. 

| British and Foreign Medical Review. 



APPENDIX I. 

be of much use, but timber, thoroughly creosoted with ten pounds of dead oil 
per cubic foot, is perfectly protected against teredo navalis. All recent au- 
thorities agree upon this point. In one instance, well authenticated, the mol- 
lusk reached the impregnated heart-wood by a hole carelessly made through 
the injected exterior. The animal pierced the heart-wood in several direc- 
tions, but turned aside from the creosoted zone. * The process and cost of 
'creosoting' have already been discussed." 

A second destroyer of submarine wooden constructions is limnoria terebrans, 
(or L. perforata, Leach) a mollusk of the family Assellotes, Leach, resembling 
the sow-bug. It pierces the hardest woods with cylindrical, perfectly smooth, 
winding holes, -^"th to ^tii of an inch in diameter, and about two inches 
deep.f From ligneous matter having been found in its viscera, some have 
concluded that the limnoria feeds on the wood, but since other mollusks of the 
same genus, Pholas, bore and destroy stone-work, the perforation may serve 
only for the animal's dwelling. The limnoria seems to prefer tender woods, 
but the hardest do not escape. Green-heart oak is the only known wood 
which is not speedily destroyed. $ At the harbor of Lowestoft, England, 
square fourteen inch piles were, in three years, eaten down to four inches 
square. § 

While all agree that no preparation, if we except dead oil, has repelled the 
limnoria, an eminent engineer has cited three cases in which that agent 
afforded no protection. | 

We do not find that timber impregnated with water-glass has been tested 
against this subtle foe. The experiment is certainly worthy of a trial. 

A mechanical protection is found in thickly studding the surface of the tim- 
ber with broad-headed iron nails. This method has proved successful. Oxy- 
dation rapidly fills the interstices between the heads, and the outside of the 
timber becomes coated with an impenetrable crust, so that the presence of the 
nails is hardly necessary. 

The following tabular statement of experimental results obtained by various 
processes is given in a report to the Board of Public Works of the District of 
Columbia, as derived from an examination of those methods by Drs. B. F. 
Craig and W. C. Tilden of the U. S. Army, in the laboratory of the Surgeon- 
general's Office at Washington. 

* Civ. Eng. Jour., vol. xii., p. 191. 

t Diet. Univ. cPEist. Natur. 

X Civ. Eng. Jour. vol. xxv., p. 206. 

§ Ibid., vol. xvi., p. 76. 

I Ibid., vol. xrv., p. 206. 

19 



EXPERIMENTAL RESULTS. 



■d 

> 

1 

83 

a 


1 

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m 

> "* ao 
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2 

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p. 


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u = a 
a "> a) 
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5 "a a 

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lbs. 
25.1 


10 

h 

"S-gS 
l?J-2 

° p. 


11 

•o 

o 
o 

1 

o 

1 

« 


1. 

8x3x6 in. 


percent 
24.29 


per cent 

1.00 


percent 

4.55 


per cent 

33.8 


percent 

1.72 


percent 

13.6 


0.418 


0.568 


lbs. 
35.5 


Acid. 


2. 

8x8x4X in- 


47.90 


0.21 


2.34 


7.7 


1.86 


6.2 


0.443 


0.472 


27.6 


29.5 


Acid. 


3. 

2x6x1 in. 


5.08 


0.43 


7.7 


5.08 


0.33 


0.4 


1.116 


1.007 


69.7 


62.9 


Acid. 


4. 
4x6x2 in. 


7.25 


0.39 


35.4 


40.00 


2.1 




0.707 


0.700 


44.1 


43.7 


Faintly 
Acid. 


5. 

4x4^x2 in. 


46.64 


1.01 


2.64 


9.55 


1.63 


7.2 


0.619 


0.6706 


38.6 


41.9 


Faintly 
Acid. 


6. 
10x6x4 in. 


3.32 


0.16 


centre& 
surface. 

17. 




0.35 




0.560S 


0.650 


35.0 


40.6 


Faintly 
Acid. 


7. 
9x4x1 in. 


8.46 


0.33 


centre& 
surface. 

22.8 




0.45 




0.845 




52.8 




Faintly 
Acid. 


8. 
6x6x8 in. 


97.58 


0.46 


1.55 


3.3 


1.46 


5.4 


0.484 


0.498 


40.2 


41.1 


Faintly 
Acid. 


9. 

7x6%x5 in. 


31.36 


0.26 


1.16 


3.45 


1.45 


1.60 


0.4609 


0.439 


28.8 


27.4 


Acid. 


10. 
12x6x3 in. 


22.80 


1.00 


1.96 


12.25 


0.52 




0.4202 


0.445 


26.2 


27.8 


Faintly 
Acid. 


11. 

12x6x4 in. 


16.15 


0.5 


14.83 


23.7 


0.38 




0.7005 


0.833 


43.7 


52.0 


Faintly 
Acid. 


12. 
12x7x4 in. 


13.54 


0.10 


16.2 


17.6 


0.29 




0.601 


0.633 


37.5 


39.5 


Faintly 
Acid. 


13. 
4x5x3 in. 


2.88 


0.13 


35.07 


41.4 


0.36 




1.095 


1.064 


68.4 


66.5 


Acid. 


14. 
6x3x2 in. 


21.27 


5.44 


31.3 


73.5 


21.9 


24.8 


0.455 




28.4 




Acid. 


15. 
8x6x3 in. 


23.19 


0.23 


10.8 


13.4 


0.52 




0.6004 


0.617 


37.5 


38.5 


Faintly 
Acid. 


16. 
10x6x3 in. 


29.11 


0.59 


1.91 


7.5 


1.8 


3.8 


0.449 


0.554 


28.0 


34.6 


Acid. 


17. 

10x6x3 in. 


31.98 


9.54 


9.17 


67.0 


6.74 


38.8 


0.449 


0.541 


28.0 


33.8 


Neutral. 


18. 
5x4x3 in. 


28.93 


0.16 










0.493 




30.8 






19. 
5x4x3 in. 


62.74 


0.64 


















Acid. 


20. 
10x6x3 in. 


78.10 


0.21 










0.429 




26.8 






21. 






1.57 


4.7 


1.63 




0.519 








22. 

Small piece 
from a pave- 
ment in N". Y. , 
laid three years 






35. 








0.407 




25.4 

(very 
dry.) 







290 



MINEEAL AND METALLIC PEOCESSES. 



Name and Nttmbeh. 



Burnettized Sprtice. 

Two specimens. Nos. 2 and 9. 
(No. 2 is a piece of railroad tie, 
Baid to have been buried sixteen 
years.) 



Brief op Claim. 



" The Burnettizing process consists in placing the wood in 
large wrought-iron cylinders ; then extracting the air and sap 
contained in the pores of the wood by a vacuum. The solution 
of chloride of zinc is then allowed to run in, and a pressure of 
from 150 to 160 lbs. per square inch applied to force the zinc 
into the pores." 

Perfect coagulation of albumen and entire indestructibility 
by wet or dry rot are claimed. 



A. B. Tripler's Arsenic Pro- 
cess. 
One specimen. No. 14. 



Saturation of blocks composing a wooden pavement with 
chloride of arsenic, or arsenic and chloride of sodium, and 
coating them on their upper surface with a resinous or tarry 
waterproof composition. Also, the interposition of an anti- 
septic compound between the blocks and the earth, by either 
soaking the foundation planks or mixing the antiseptic with 
the sand. 



The Samuel's Process. 
Nos. 8 and 5 (?). 



" Injecting into the pores of the wood, first, a solution of 
sulphate of iron, and afterwards a solution of common burnt 
lime, to render the wood in a high degree impervious to the 
influence of wet and dry rot, and the attacks of worms and other 
insects." 



Thilmany's Process. 

One specimen. No. 16. 



Saturation with sulphate of copper, followed by muriate of 
barytes, to form insoluble sulphate of barytes in the wood. 



Process of Wirt and Hurdle. 

Specimens 18 (a) and 18 (b). 



Charring the wood and covering the whole block with as- 
phaltum. 



Tait's Process. 

One specimen said to have 
been sent, marked No. 5. Anal- 
ysis, however, places this block 
with No. 8, as a sample of the 
Samuel's process. 



" Charging or saturating the pores of the wood with a con- 
centrated solution of bi-sulphite of lime or baryta, the same 
being rendered soluble by excess of sulphuric acid gas, under 
pressure or by refrigeration, and being made insoluble as a neu- 
tral sulphate when the pressure or excess of gas is removed. 1 ' 



Thomas Taylor's Process. 

Two specimens. Nos. 19 & 21. 



Uses a solution of sulphide of calcium in pyroligneous acid 
for the impregnation of the wood ; or, uses sulphide of calcium 
first, and follows it with pyroligneous acid. 

Claims a deposit of pure sulphur through whole block. 



Thompson & Co.'s Process. 

v Arsenic.) 
No. 17. 



No description or explanation of process furnished. 
Claims "indestructibility" and "non-inflammability. 



291 



Laboratory Surgeon-General's Office. 



Remarks. 

1~ ' 



Fibre of blocks weak and brittle ; color grayish. 

Absorptive power greater than that of natural wood. All of the zinc easily removed by 
acidulated water. Evidences of the partial decomposition of the zinc chloride observed. Uneven 
character of impregnation shown both by microscopic examination and by unequal percentage of 
mineral matters removed by acidulated water from centre and near surface. (See Columns 3 and 
4 of Experimental Results.) 



Size of specimen very small, yet the impregnation uneven. (See Columns 3 and 1 ) Quantities 
of soluble salts very large. No arsenic found, though its use is claimed. The resinous covering 
designed to protect the top of each block is worthless for the purpose, for obvious reasons, chiefly 
its brittleness. 

Absorptive power high. 



Absorptive power too high for representation on the chart. "Wood brittle and readily splintered. 
Impregnation very unequal. The water used for Experiment No. 1 (absorptive), was filled with 
threads of fungi after standing forty-eight hours, showing that it is doubtful if even dry rot can 
be prevented by this process. 



Saturation very uneven. Absorptive power high. 

Block contains soluble salts of copper removable by washing. 



Process inapplicable to unseasoned timber. The asphalt covering melts and flows at 60° to 
70° P. When cold and brittle, the wear of the pavement will remove it, leaving each block as a 
poroiis cup for the reception of water which cannot drain through it. Process not considered 
worth particular investigation. 



It is doubtful if any specimen was received. No. 5 resembles the "ironized" blocks. If 
claimed as a sample of the Tait process, the same memoranda are made upon it as upon No. 8. 



The claims of this process are not substantiated. 
No sulphur uncombined found in any part of blocks submitted. 

About nine-tenths the whole bulk of each block possessed every property of seasoned white 
pine untreated by any method whatever. 
Between three and four per cent, sulphate of lime found in superficial portions. 



An arsenic process. Absorption power high. Specimen is cottonwood. 
Saturation extremely uneven. Solubility of saline ingredients complete. 



292 



CEEOSOTE OIL AND KESIN PROCESS. 

Laboratory Surgeon-General's Office. 



Name and Numbeb. 



Bbief of Claim. 



Waterbury's Pro- 
cess. 

One Specimen. 
No. 1. 



Thomas' Process. 

Two Specimens. 
Nos. 3 and 13. 



Treats wood in closed cylinder 
with steam to vaporize sap ; then 
introduces a solution of common 
salt, followed by dead oil, creosote 
oil, or equivalent. Claims com- 
plete impregnation by both sub- 



Two small blocks, 2x6x1 and 
4x5x3 in., were sent without ex- 
planation or name : the substance 
used for impregnation is "resin 
oil." 



EiEMABKS. 



Absorption figures high. Satu- 
ration by solution of common salt 
is only partial. Columns 3 and 4 
show a very uneven penetration 
by '■'•dead oiV Water dissolves 
out all the salt used. Columns 5 
and 6 show the uneven distribu- 
tion of mineral matters. 



Absorption power low. Physical 
condition of specimens very bad. 
Saturating material easily soluble 
in alkaline fluids. The strength of 
wood in these samples stands at a 
minimum, especially its transverse 
and crushing strength. 



Seely's Process. 

Five Specimens. 



Pelton's Apparatus 

for applying Seely 

Process. 
Nos. 4, 6, 12, 22. 



Bobbins' Process. 

Two Specimens. 
Nos. 10 and 20. 



Detwiler and Van 
Gilder Process, 

No. 11. 



TJ. S. Antiseptic 
Wood Co.'s Process. 

Constant and Smith 
Patents. 
No. 15. 



Immersion of wood in a bath 
of creosote oil or other suitable 
material, heated to about 250° F., 
until it is evident that air and 
moisture are eliminated ; then 
substituting for the hot bath, one 
at as low a temperature as allows 
perfect fluidity, the liquor being 
also dead oil. Claims that the 
pores of the wood are in a vacu- 
ous condition as it cools, and that 
the impregnating material readily 
fills them by capillary action and 
atmospheric pressure. 



Average absorption power very 
low. Saturation thorough and very 
uniform. (See Columns 3, 4, 9, and 
10.) Solid hydrocarbons present 
within the cells. Condition of 
fibre uninjured. 



Claims to impregnate wood 
with light and heavy oils of tar, 
by exposure in a chamber con- 
nected with a retort or still in 
which the oils are vaporized ; 
states that naphthalin and other 
solid hydrocarbon bodies are dis- 
tilled over into the wood and con- 
densed in its pores ; also that all 
moisture is driven out and the 
albumen coagulated. 



Absorption power very high. 
Percentage of liquid hydrocarbons 
very low in all portions of block 
except the outer. No solid hydro- 
carbons observed, even on surface 
(naphthalin, etc.). Condition of 
wood shows injury from heat 
Specimens are evidently suited foi 
exposure to dry air only, under 
which circumstances the protection 
is sufficient. 



Impregnation of wood by resin 
dissolved in naphtha, under pres- 
sure, and at high temperature. 



Saturation uneven. (Columns 3 
and 4, also 9 and 10. ) Absorption 
power quite high. 



Dries or seasons wood by hot air; 
preserves it (when desired) by 
generating " smoky vapors " in a 
retort, the same being allowed to 
penetrate the wood and to con- 
dense within its pores. 



The same remarks made under 
Nos. 10 and 20 (Bobbins' process) 
apply to this specimen, with the 
difference that the experimental 
results show the Bobbins' process 
to be very much superior to this 
which presents identical claims. 



293 



APPENDIX II. 



(The numbers of the articles correspond with those in the text.) 

19. To integrate -=75- = P j- x\ multiply both members by dx, and the 

dxd 2 x 
first member becomes — ^»- . Let dx = s, then d 2 x = dz, and the expression 
Cut 

becomes -^. t being- the independent variable, dt is constant, and hence the 
at 

z 2 dx 2 
integral is ^3-5 = z-=-^. The other reductions are evident. 
2dt 2 %dt 2 

The resilience (or spring of the bar) is the work of elongating it to the limit 

of proof strain. Equation (7) of the text gives 

EKk* _EK PH 2 P 2 
U ~ 21 *~. »"" E 2 K 2 ~ EK 2 ' * KL 

p 2 
P+ Kis the stress on a unit of surface. The quantity ^ 2 , is called the 

modulus of resilience. 

22. The strength of metals is referred to the original section, but before 
rupture takes place the section is considerably reduced. See the table in Ar- 
ticle 35, and the example in Fig. 116. In some cases it may be reduced to J 
of its original section before it ruptures. 

32. At the bottom of p. 40 there is a remark which, in the light of more 
recent experiments, is probably incorrect. The probable effect of the tension 
was to relieve the iron of internal strains, and by leaving it for a time without 
strain, and permitting it to cool, all the elements may come to a condition 
of maximum resistance. 

96. The moment of inertia of a surface is the sum of all the products ob- 
tained by multiplying each elementary area by the square of its distance from 
an axis. 

The moment of inertia cf a volume is defined in a similar way. Similarly 
for a weight, or for a mass. 



APPENDIX. 



m* 



MOMENT OF INERTIA OF A SURFACE. 

Let J = the moment of inertia about an axis ; 
dA = the elementary area ; and 
y == an ordinate from the axis to the element. 
Then 

I=J*y*dA 

If the ayes are rectangular d A : 



.fl> 



—jj V 2 dydx 



Example.— Required the moment of inertia of a rectangle when the axis 
ooincides with one end. 
Here 



~J «/o 



y^dydx = %bd* 



FORMULA OF REDUCTION. 



# 



Fig. 123. 



Let J = the moment of inertia about an axis through the centra ; 
J = the moment about a parallel axis ; 
D = the distance between the axes (Fig. 836) ; and 
y = the ordinate from the axis through the centre to an element. 

Then 1= fy 2 dA =J*(y - Df dA = fy\ dA - pZDy Q dA+J*D 2 dA 

= J + I> 2 A, 
for 2D I y dA = since the axis passes through the centre. (See foot note, 

p. 88.) 

That is, the moment of inertia about any axis equals the moment about a par- 
allel axis through the centre, plus the area of the section multiplied by the square 
of the distances between the axes. 

From the preceding equation we have 

J =I-AD* (3) 

Example.— Required the moment of inertia of a rectangle about an axis through tie centre and 
parallel to one end. 
Here A — bd, D 2 = \<P, and I = \bd* as found above ; hence 

7 = fyd 3 — \bd* = -£zbd* : 
as given in Eq. (51) of the text. 

If the moment axis is perpendicular to the surface, we let 

p = the variable distance of any element from the axis ; 
<j> = the variable angle; 
dA = pdcpdp (see Fig. 78) ; and 
Ip w the polar moment of inertia. 



:I p = fp*dA 



296 APPENDIX. 

But p* = x 2 + y* 

.'.I p =Jx 2 dA+Jy 2 dA = Iy + l x (4) 

in which I y is the moment about y and l x the moment about x. If lx — ly 
we have 



The polar moment of inertia of a circle is Eq. (199) of the text, 
I p = iTrr 4 = 27* 
.'.Ix = i*T 4 (5) 

which is the moment about a diameter, as given in Eq. (52). 

TO FIND THE BELATION BETWEEN THE MOMENTS OP INERTIA ABOUT 
DIFFERENT AXIS HAVING THE SAME ORIGIN. 




Fig. 124. 

Let x and y be rectangular axes, 

Xi and y u also rectangular, having the same origin; 
. a = the angle between x and x x ; 

lx = the moment of inertia about the axis #, similarly for 
Iy, lx-i and Iy x ; 



B= lxydA\ 
Bx =Jx 1 y 1 dA. 



and 



For the transformation of coordinates, we have 

x x = x cos a 4- y sin o 
y x = — x sin a + y cos a 



Also 



dA = 



Hence, 

J XI = J y\ dA = lx cos 2 a + I y sin 2 a — 25 cos a sin a | 

Iyi = ia sin 2 a + Iy cos 2 a + 25 cos a sin a V .. . ,(fl) 

I?] = (I x — Iy) cos a sin a +5 (cos 2 a — sin 2 a) 
• °. lx\ + ly — 1st + Iy "= Ip J 



APPENDIX. 



297 



which is an isotropic function ; since the sum of the moments relatively to a 
pair of rectangular axes equals the sum of the moments relatively to any other 
two rectangular axes having the some origin ; or, in other words, the sum of 
the moments of inertia relatively to a pair of rectangular axes is constant. 

From the first of (6) we have by differentiation and placing equal to zero 
for a maximum or minimum, 



— -~ l = (I x — ly) cos a sin a +Z? (cos 2 a — sin 2 a) = 

Q/CL 



Similarly, 



-~i = (J x — ly) cos a sin a — B (cos 2 a — sin 2 a) = 



.'. B x = 0. 

From the first or second of these we have 



— 2B 



2 cos a sin a 
cos 2 a — sin 2 i 



= tan 2a. 



It may be shown by the ordinary tests that when I x x is a maximum, that 
ly j will be a minimum, and the reverse ; hence there is always a pair of rec- 
angvXar axes in reference to one of which the moment of inertia is greater than 
for any other axis, and for the other it is less. 

These are called principal axes. y 

Thus, in case of a rectangle, if the axes are parallel to 
the sides and pass through the centre, we find 



-If. 



+\d 

xydA = ; 
U 



Fig. 125. 



hence x and y are the maximum and minimum axes ; and 
if d > b, -febd* is the maximum, and -feb^d a minimum mo- 
ment of inertia for all axes passing through the origin. In 
a similar way we find that if the origin be at any other point 
the axes must be parallel to the sides for maximum and minimum moments. 

The preceding analysis gives the position of the axes for maximum and min- 
imum moments, when the moments are known in reference to a pair of rec- 
tangular axes. But if the axes for maximum and minimum moments are 
known as l x and I y , then B — ; and calling these I x ' and I y ', and Eqs. (6) 
become 

lx i = Ix cos 2 a + I y ' sin 2 a ) 

Iy i = Ix sin 2 a + Iy' cos 2 a V (7) 

2?! = (I x ' — Iy ') cos a sin a ) 

In the case of a square where the axes pass through the centre I x ' = Iy' 

,\Ixi = Ix' (cos 2 a + sin 2 a) = I x ' 
Iy j = Iy' , and 

hence the moment of inertia of a square is the same about all axes passing 



298 



APPENDIX. 



through its centre, and = -feb*. The same is true for all regular polygons, 
and hence for the circle. 

Example* — To find the moment of inertia of a rectangle in reference to an axis which is in- 
clined at an angle a to one side, 

We have IJ = -febd*, I J - -fabU ; 

.'. J„ = -febd (& cos* a + 62 B inS a ) ; 
I n = -febd (d 2 sin 8 a + 62 cos 3 aj. 

The moment of inertia of a regular polygon may be found by dividing it 
into equal triangles, having their vertices at the centre, their bases being 
sides of the polygon, and finding the moments of each in reference to an axis 
passing through their centre, and parallel to the axis which passes through 
the centre of the polygon, then reducing all of them in reference to the main 
axis. If R be the radius of the circumscribed circle, and r the radius of the 
inscribed circle, it may be found for any regular polygon, that 

I=^A{R' +2r*) (8) 

For the circle R = r 

as before found. 
For the square r = lb, R = 

as before found. 



^W=bY2,&ndA = b 2 i 



96. The value of p (the radius of curvature) given in the latter part of this 
article, is taken directly from the Differential Calculus, but it may be easily 
found geometrically as follows : — 




Fig. 126. 

From two consecutive points erect normals, and where they meet at will 
be the centre of the oscillatory circle, and the normals will be sensibly equal, 
and will be the radius of curvature p. Let d<p be the angle between the nor- 
mals ; ds = the arc — pd$. 



The coordinates of a in ref erenoe to b will be dx and 
pendicular be to 0^ and the angle abc will equal d<p. 



At b erect a per- 



APPENDIX. 299 



Take the differential of this and we have 

d 2 y _ di d<j> 

dx ~ cos 2 i ~~ dx z 

ds 2 

'^ = dt = -d^=J 

- _^ 3 _ _ (dx 2 + dy 2 ) $ 
''* P ~d 2 ydx~ d 2 ydx 

d 2 y 
99. For the integral of the expression -~ see Article 19 cf this Appendix. 

dx 

114. Equations (130) and (131) are particular cases of the more general 
form 

3^£+*-*=/« » 

which may be integrated by La Grange's method of variation of parameters. 
(See Price's Infinitesimal Calculus, 1st Ed., Vol. II., pp. 479 to 481.) 
First let X = 0, then 

2 + *!+*=• •• • 

Let y = e mx in which m is constant and e the base of the Naperian system 
of logarithms ; then 

^=™*™*and^ = ™ 2 * ma! (3) 

ax ax 

which in Eq. (2) becomes 

m 2 + Am+ B=0 (4) 

which is an equation of condition. 
Solving gives 

m = \A± «\A 2 -B=a and b (say) (5) 

where a and b are the roots of Eq. (4). Hence the partial values of the first 
differential coefficient are 



(£)——(£)=»■* 



In order that these should satisfy Eq. (1), a and b must be functions of X. 
Integrating each and introducing u x and u 2 , which arc functions of x instead 
Df arbitrary constants, and we have 

(y) = TJ x <fl* and (y) = U 2 #* 

.\y = U x <px + TJ % #* (6) 



500 APPENDIX. J 

It now remains to find Ui and JJ 2 . Differentiate Eq. (6) and we find 

*=V ia *. + U.». + *.™l + +.*£. (7, 

But such a relation may be established between U x and JJ 2 as that the sum 
of the last two terms shall be zero, or 

,*.*£+*.**! = (8) 

dx dx 

which is a second equation of condition. Hence Eq. (7) becomes 

^= U x a#* + U 2 b(P* (9) 

CLX 

Differentiate again : — 

= U x a 2 ^ + U 2 b 2 e»* + a<P*^ + bJ>*^... .(10) 

Substitute Equations (6), (9), and (10) in (1) and we have 

UAa z + aA+B)e™ + U 2 (b 2 + bA+B)<P* +a^^ + b^^ = X. 

dx dx 

But the first equation of condition, Eq. (4), reduces the first two terms to zero 

dx dx 

which, combined with Eq. (8), eliminating dU ly and then dU 2 , and we find 

U 1 = d + -i- x fxe-«x dx, 
a — b*J 

^2= G 2 - ~-g CxiH>* dx, 

which substituted in Eq. (6) gives 

y =C l &** + C 2 e b * + / Xe-™ das — -— x / Xe~ bx dx. .(11) 

a — oj a — b J 

By comparing Eq. (130) of Article 114 with Eq. (1) above gives 

4 = 0; B=-q*aa&X=—p*x; 
hence Eq. (4) gives 

m— ± q .*. a = + q and 5 = — q; 
which reduces Eq. (11) to 

y= dew + C 2 er-Q*-^~ xe-<&dx + z-^ — lx&*dx. 



To integrate ixe-Q x dx first differentiate xe-Q* , 



APPENDIX. 301 

.'. d (xe-Q x ) = —qxe-w dx + e-Q* dx, 
from which we find 

xe-Q* dx = —e-< x dx d (xe-Q* ) 

or jxe-w dx = - /Y-fl* dx f d (xe-Q x ) 

The integral of / e—Q x dx is e-w* and the last term is the integral of 

the differential of a quantity, hence 

/l 1 1 
xe~Q x dx = 5 e~Q x xer<& = ^e—^ (1 -+- gx) 
q> q q 2 

.•.^_y ia ^ < te=_| i (i + sa ,). 

p 2 
Similarly the last term becomes — ^— ^ (1 — qx) ; and hence the former sub- 

P 2 
tracted from the latter gives — 2 x. 



.-.y= dee* + de-* x + — % x. 

The following special solution, which is more simple for this case, is com- 
municated by Prof. S. W. Robinson, of the Industrial University of Illinois. 
To integrate 



Differentiate twice and 



dx* ~ * dv£ 



d 2 v d^v d 2 u d 2 v 

Put -r-f = wand differentiate twice and —i — -— = q 2 ~ 

dx 2 dx 4 dx 2 * dx 2 

, d 2 u . _ dud 2 u 

r^= /^ = " l0S(aV ^ ) " l0^<7,, 

or &* = -zz 



302 



APPENDIX. 



C l efP i — u = a/u*+ — , square, smdO^e 2 ^ -W x e<& u +« 2 = u*+ £. 
V g z g* 



u = -k — 



2(7 1? 2 
Bvt^ = u = q*y-p*x = - ^ 

Or C P z 



which is the same as in text by putting 



Equation (131) gives 

A = ; B = q 2 . • . m = ±q \T^1 .'. a = q 4/— 1 and b ~ — q 4/— 1 and 

X = p* (I - x). 

Make A — y = — y', which in Eq. (11) above gives 

-y' = C\ea*tf-i + ^r^r 1 ,-^ (l-as)- A -y. 

But we have (Chauvenet's Trigonometry, p. 128, Eq. (432) ), 
esx 4/— 1 — cos qx + \/—l sin qx 
e-Qx 4/— 1 = cos qx — 4/— 1 sin qx. 
Make (? x + (7 2 = J-i and (C7 X + (7 2 ) 4/^! = #1 and we find 

^ — y = ^.j cos qx + #! sin ga? — ^ (Z — a>). 

Let A x = Asinq B 
Bi = Acob q B 

.• . A — y = A sin q (x + B) — ^ (? — aj). 

115. Suppose that the column is fixed at the lower end and has a weight 
P at the upper end. Take the origin at the lowest end, and let A be the de- 
viation of the upper end from a vertical. Then 

The complete integral gives 

y = J + A cos \/jj}j + BsmyjjjjX, 
in which A and B are the constants of integraticn. 



APPENDIX. 



303 



y = for x = and -~ = f or x = 

.*. A = — J and 5= 



= Jfox x =£ Z .* . cos \/ -=j i or 

.-.^/Ji = *<*» + !)« 



in which 71 is an integer. 



If» = 
P = 



4Z 2 



The general equation of the curve is 
V = A (l - 



2rc + l > 
cos — pi — irx 



"1 

Fig. 127. 



Fig. 128. 



If 7i = 2, the curve is represented by Fig. 128. 

A solution was given by Prof. Rankine in the Civ. Eng. and Arch. Jour. 
for 1863, p. 65, by which the deflection may be computed. 

1 46. See Article 96 of Appendix. 

To find the inclination so as to give a minimum strength, make the first 
differential coefficient of Eq. (172) equal zero. This gives 

= (d* — 2b 2 d) sin 2 i cos i + (2bd 2 — 5 3 ) sin i cos 2 i + bd 2 sin 3 i — b 2 d cos 3 i. 

But sin 3 i = sin i (1 — cos 2 i) and cos 3 i — cos i (1 — sin 2 i) which will re- 
duce the preceding equation to 

(d 3 — b 2 d) sin 3 i cos % + {bd 2 5 3 ) sin a* cos 2 i + 5tf 2 sin »" — b 2 d cos & = 0. 



In this substitute sin i = vl — cos 2 * and reduce, and we obtain 



304: APPENDIX. 

[_ {pa 2 - b 3 ) 2 - (d 3 - b 2 d) 2 ] cos 6 *• + [(bd 2 - b 3 ) 2 - 2 (bd 3 - b 3 ) bd 2 - 2 
(2b 2 d - d 3 ) (b 3 - b 2 d)1 cos 4 i + [2 (bd 2 - b 3 ) bd 2 - b 2 d* - (2b 2 d - a 3 ) 2 ] 
cos 2 a = - b 2 d* (1) 

Discussion of Equation (1). 

1. Let d = nb ; then we have 

(n 6 — 7i 4 — n 2 + 1) cos 6 i — (2n 6 — 7 n* + 4 ft 2 + 1) cos 4 i + (n 6 — 5 n* + 
6 n 2 ) cos 2 »' = ra 4 ; 

hence the angle depends only upon the ratio of the sides. 

2. Let n = 1. 

This reduces to cos 2 i = -£. 

. •. cos i s= + Vi- •'• * = 45° or 135°. 

These values give 

Rb % 
£— = £ i$ 3 x 0.70710 +, for the strength of the beam. 

Ef b = d and i = 90°, we find ^ Rb 3 for the strength of the beam. 

Hence the strength of a square beam with its side vertical, is to the strength 
of the same beam with its diagonal vertical as 1 to 0.70710. But if the con- 
dition be that the beam shall in both cases be completely severed, then the 
latter fraction must be multiplied by 1.09125 +, as shown in Article 149. 
Then the ratio becomes as 1 to 0.77162 -K 

3. Let n = 2 and cos 2 i = y. 
Then the equation becomes 

y*-Uy* + -hy = U (274). 

To make the second term disappear, make y = z + \\, and it becomes 

3 31102 



■z — 



45 3 " 45 3 

This solved gives z = 0.70112 

.-. y = 0.94556 + = cos 2 i 
.'. cos£ = ± 0.9723 + 
.-.*= 13° 30' or 166° 30' 

making d == 25, and we have for the strength of the beam 

when i = 13° 30' ; %Rb 3 x 0.8295 + 

" *'= 0° ; \Rb 3 

" * = 90° ; %Rb 3 . 

It is probable that in the inclined position the angle would fracture before 
the beam is loaded to its ultimate strength, but the investigation for determin- 
ing it would_ be tedious and unprofitable. "Whether this be the case or not, we 
see that the beam is not weakest when it rests on its broad side. 

It appears that the side of the beam may be so inclined as to have the same 



APPENDIX. 305 

strength as when it rests on its broad side, and the angle of inclination which 
will fulfil this condition may be found by making Eq. (1) equal \Eb 2 d. 
This done, cos i eliminated, and a reduction made, gives ; 

(d 2 — b 2 ) 2 sin 3 i-2 {d 2 - b 2 ) bd sin 2 i + {Zb 2 d 2 — 6 4 ) sin i = 2b 3 d. 

Ifd = nb, we have 

{n 2 - l) 2 sin 3 i-2 (n 2 - 1) n sin 2 i + (3ra 2 - 1) sin *' = 2n. 

If n = 2 we have 

. 3 . 4 . . . 11 . . 4 
Bin 3 I — * sin 2 i + — sin a = -jr 

which solved gives i — 34° 23'. 

149. The moment of inertia of the trapezoid DEAB in reference to an axis 
passing through C (Fig. 77) and parallel to IJ, equals the moment of inertia 
of the triangle ABC, less that of CDE about the same axis ; or \ bd 3 —$vw 2 . 

According to the formula of reduction, the moment of the trapezoid in 
reference to the axis IJ, equals the moment given above, less the area of the 
trapezoid multiplied by the square of the distance GH; or 

^(bd z -'ow z )-\{b+t>) (di-hw)* (d-w). 
To find ri?i, we have the statical moment of ABBE equal to the statical 
moment of ABC, less that of CDE. Take the origin of moments at C7, and 
we have 

ABED x {w+d x )=ABCx §d-DCEx $w 
or, i(d-w) (b+v) (w+dJ^bd'-bQw*. 

We also have 

d 

W=r'D 


whence by elimination and reduction we find the expressions in the text. 

150. To integrate r* sin 2 <fd<j>dr substitute i (1 — cos 2 <p) for sin 2 <p, which 
gives 

ffr* (1 — cos 2(f) drdb = JrV — -Jr 4 sin 2<p. 

151. If a = semi-transverse axis 

b ~ semi-conjugate axis 
The equation of the ellipse is 

a 2 y 2 + b 2 x 2 = a 2 b 2 



.•./=/ / y 2 dydx = *- y*dx = ?L. / (b 2 - x*)*dx, 
J—yJ—a SJ—a 3a V— a 

which, by applying formula (J5), p. 285 of Courtenay's Calculus, becomes 
which for the limits x = and x = a becomes Inab 3 . 



306 



APPENDIX. 



A a 



5 




m 



b 

Fig. 129. 



152. The origin being at the vertex, x horizontal, 

the equation for the curve is of the form x* = %py ; or 

b 2 
for the point B, ib 2 = 2pd . • . 2p = — ; hence the equa- 

b 2 4:d 

tion is x 2 = j- y . *. y = — x 2 ; hence by substituting 

the value of / and integrating we have 



: I / y 2 dydx = 2 1 y 2 xdy = A / yidy =$bd* 



Or thus : 
1 = 



JJy 2 dydx =£ ly*dx. The part £y 3 d£ is the moment of ah, but we 

want the part he, which is within the parabola ; hence the limits for y are 
d — ac and y = ali\ and for x ; ib and — -£6, 

A J / y*dy<fe = i / (<* 3 - y 8 ) <fe = i / (d* - 6 ^-x«\dx = W 3 

as before. 

To find the moment in reference to the axis passing through the centre and 
parallel to the base, we may use the formula of reduction and obtain 



b& 



E bdx 



(J')"- 



8 



bd* 



7 ~- 3 

160. a. Expand and we have 

6xj ydx — 6 Jxydx ±= \By 2 

Consider x as the independent variable, and differentiate twice, observing 
that the differential of dx will be zero, and the differential of the integral is 
the original quantity, and we obtain 



r b 

dxjydx = -Q- 6 d{y 2 ), and 



d 2 (y 2 ) _ 6-f 
dx 2 ~ B y 

Let y 2 = z . *. y = s* and d 2 {y 2 ) — d 2 z, which, substituted in the preceding, 
gives 

dx 2 ~ B*' 

Multiply by dz and integrate, and we have 

dz 2 8S a n 
cW = B S2 + G > 



APPENDIX. 307 



But ? = for y ^ ors = .♦. O x = 0. 

(IXC 



Hence 







Z' 


-*<& = 


V?" 




of which the integral is 














4*-: 


= 4pi = 


VI- 


C s 


Buty = 


= f or x = . 


'.0, 


= 0, 


= 33?*'- 





If — is not zero for y — 0, it can be integrated only by Elliptic Functions, 
dx 

b. By two differentiations we obtain 

Rd 



The first integral is 



1st, tidx judx = -g- <?«. 

nJ rf 2 « 6(5 
2d, -=-2-= ^r M - 
die 2 i?d 

■■■■= Vf^C/f^^--]^' 

The curve will be an asymptote to the axis of x, and hence will be of infi- 
nite length. If at the origin u = 1 for x = 0, the value of G ' may be deter-* 

rained. Similarly if — = a at the origin (7 may be determined, 
c. By expanding we obtain 

6x.jy 2 dx — djy^xdx = £i?y 3 . 
The first differential is 

Mxfy 2 dx = ±M (y 3 ). 
The second differential coefficient is 

Let y* —z and the integration may be performed as in the preceding cases ; 
the constants of integration being considered zero. 



308 APPENDIX. 

179. Differentiate Equation (213) considering IT as constant, and we hare 

dy 1 y 

This expression has been generalized. Consider g' as the amount of trans- 
verse shearing per unit in the direction of U, and G' the coefficient for the 
same. Then we have 



dg dg' EP 

°dy+ G dy = -r< 



dx *._ dgd*x 



Kef erring g' and g to the co-ordinate axes, if g = — , then, — =x . 

dy dy dy* ' 

and we have 

„ c^a? „, tffe LP 

On page 216 it is remarked that Eq. (213) is not generally exact. Conceive 
that the transverse section is an ellipse. Conceive also that it is divided into 
several vertical strips. If the longitudinal shearing increases from the sur- 
face to the neutral axis, then will there be a greater shearing strain on those 
strips near the middle than on the outer ones. Hence the tranverse sections 
will not be cylindrical (having for base a curve whose Eq. is (217a) ), but they 
will be warped or generally distorted. This is a refinement, however, which 
it is not necessary to consider in practice. 

y 2 2P 

On the compressed side of the beam, the quantity - (p. 228) should be 

subtracted from g in finding the equation of the curve. 

190. Crystallization. — Several illustrations of apparently unmistak- 
able crystallization have recently been brought to the attention of the author. 

In the process of forging a large shaft for a sea-going steamer, the steam 
nammer was at work upon one end, as in the case of the ' ' porter bar " already 
referred to in the text, and while the mass was gradually being reduced in 
size, a piece of the opposite end broke off at a point where it was 10 inches in 
diameter. The fracture was partly granular and partly crystalline. One 
crystal was unmistakably cubic, and its facets were nearly a half -inch square. 

At the Washington Navy Yard, recently, the large testing machine, which 
was constructed for a strain of 300 tons, broke down under a pull of 100 tons. 
The rod which was broken had been in use 35 years, and, during that period, 
had been subjected to many heavy strains and violent shocks. Not long be- 
fore it was broken it had been subjected to a tension of 288,000 pounds. It 
was 5 inches in diameter, and was originally supposed to have had a resistance 
of about 1,000,000 pounds. The fracture presents a granular structure, with 
here and there laminae composed of crystals. Some of these crystals are large 
and well defined. The laminae or strata preserves their characteristic pecu 
liarities, whether of granulation or of crystallization, lying parallel to the axis 
and extending from the point of original fracture to a section about a foot 
distant where the bar was broken a second time by a steam hammer. It is 
thus shown to be the fact that when such true crystallization does occur, it 
pervades a considerable extent, if not the whole of the piece. It thus differs 
from the granular structure which distinguishes the surfaces of a fracture 
suddenly produced, and which is so generally confounded with rca 1 crystallis- 
ation. 



APPENDIX. 309 

The above instance is given by Prof. Thurston, who also describes the fol- 
lowing case. A pupil of the Stevens' Institute of Technology, employed in 
the instrument makers' workshop, in annealing a number of steel hammers, 
left them exposed to the high temperature of the furnace about twelve hours. 
When finishing one of them, a careless blow broke it, and the fractuied sur- 
face was found to have a distinctly crystalline character. 

In this example, however, the facets were all pentagonal and were usually 
very perfectly formed. 

;t These illustrations,'' our informant remarks, " are conclusive of the ques- 
tion whether iron may crystallize. When imperfect cubic crystals are devel- 
oped, it is easy to mistake them, but the formation of pentagonal dodecahe- 
dra, in large numbers and in perfectly accurate forms, may be considered 
unmistakable evidence of the fact that iron may crystallize in the cubic or a 
modified system. This may apparently take place either by very long contin- 
ued jarring, or under the action of high temperature, by either mechanical 
or physical tremor long continued. But no evidence is given here that a sin- 
gle suddenly applied force producing fracture can cause such a systematic and 
complete rearrangement of molecules. The granular fracture produced by 
sudden breaking, and the crystalline structure produced as above during long 
periods of time, are as distinct in nature as they are in their causes. 

' ' But simple tremor, where no sets of particles are separated so far as to ex- 
ceed the elastic range, and to pass beyond the limit of elasticity, does not seem 
to produce this effect, however. 

" In fact, some of the most striking illustrations of the improvement in the 
quality of wrought -iron with time have occurred where severe jarring and 
tremor was common. As one example, the first wrought-iron T-rails ever 
made were laid down on the Camden and Amboy Railroad in 1832. They 
were then brittle and of decidedly poor quality. In later years these old rails 
have been taken up and found to be of excellent quality ; and when there has 
arisen a necessity for a supply of unusually good iron, a lot of these rails has 
sometimes been taken up and sent to the rolling mill to be made into bar 
iron. 

" Here the metal has been subjected for many years to the strains and 
tremor accompanying the passage of trains without apparent tendency to crys- 
tallization, and with evident improvement in its quality. The fact is stated 
by gentlemen upon whom perfect reliance may be placed. The improvement 
noticed is supposed to be due to a surface oxidation of the injurious elements 
originally present in the iron, and to that tendency to uniform diffusion which 
gradually supplies new portions from the interior, until the metal, by this grad- 
ual removal of those elements, becomes, after many years, comparatively pure. 
Such a process of diffusion occurs, in the inverse direction, when carbon is 
introduced into steel by the cementation process. Many illustrations of this 
improvement of metal with age are familiar to every mechanic. Boiler- 
makers find chisels and drift- pins which are taken from boilers where they 
may have been left months previously, almost invariably of excellent quality ; 
other mechanics find that tools long lost and again found rusty from exposure 
to the weather, are apparently of better quality than before ; farmers leave 
their new scythe blades out of doors from one season to another with a result- 
ing benefit which is not all imaginary." 



APPENDIX II. 

TABLE 

Of the Mechanical Properties of the Materials of Construction. 



Note. — The capitals affixed to the numbers in this table refer to the following authorities :- 

B. Barlow. Report of the Commissioners of 
the Navy, etc. 

Be. Bevan. 

Bn. Buchanan. 

Br. Belidor, Arch. Hydr. 

Bru. Brunei. 

C. Couch. 
CI. Clark. 

D. Darcel, Annales for 1858. 

D. W. Daniell and Wheatstone. Report on 
the stone for the Houses of Parliament. 

E. Eads. 

F. Fairbairn. 
G-. Grant. 

H. Hodgkinson. Report to the British Asso- 
ciation of Science, etc. 

Ha. Haswell. Eng. and Mech. Pocket-Book, 
1869. 

J. Journal of Franklin Institute, vol. XIX., 
p. 451. 

K. Kirwan. 



Ki. 


Kirkeldy. 


La. 


Lame. 


M. 


Mischembroeck. Introd. ad Phil. Nat. L 


Ma. 


Mallet. 


Mi. 


Mitis. 


Mt. 


Mushet. 


Pa. 


Colonel Pasley. 


R. 


Roudelet. L'Art de Batir, IV. 


Ro. 


Roebling. 


Re. 


Rennie. Phila. Trans., etc. 


S. 


Styffe. On Iron and Steel. 


T. 


Thompson. 


Te. 


Telford. 


Tr. 

Iron. 
W. 


Tredgold. Essay on the Strength of Cask 


"Watson. 


Wa 


Major Wade. 


Wn 


. "Wilkinson. 



* Calculated from the experiments of Fair- 
bairn and Hodgkinson. 



Names of Materials. 


11 

If 


i 

u 

15 


Crushing Force 
per square inch 
in lbs. 

C. 


3U 

'a cu 
o s 


OH 


Metals. 

Antimony — 

Cast 


281.25 
613.87 

525.00 
534.00 

537.93 
549.06 
560.00 


1,066 M. 
3,250 M. 

17,968 Re. 


10,304 Re. 










Brass — 


9,170,000 
14,230,000 






Copper — 

Cast 


19,072 
32,184 
61.228 
48,000 


29,272 Re. 


48,240 T. 

38,556 H. 
87,503 H. 

35,f)80 F.* 
42,687 F* 


Sheet 








In Bolts 




Iron. 

Cast Iron. 

Old Park 




18,014,400 T. 
17 270 500 H.» 


Carron, No. 2 — * 

Cold Blast 


441.62 

440.37 

443.37 
441.00 


16,683 H. 
13,505 H. 

14,200 H. 
17,775 H. 


106,375 H. 
108,540 H. 

115,442 H. 
133,440 H. 


Hot Blast 


16^08o'000 H.* 
16 246 900 F 


Carron, No. 3 — 

Cold Blast 


Hot Blast 


17,873,100 F. 



APPENDIX. 
TABLE.— Continued, 



311 



Names of Mateeials. 






£B 



P< CO 

•1.a 



H.S 



P' 






g£ri 



Iron. 
Cas« Jrcm. 

Devon, No. 3 — 

Cold Blast 

Hot Blast 

Buffery, No. 1— 

Cold Blast 

Hot Blast 

Coed Talon, No. 2— 

Cold Blast 

Hot Blast 

Elsicar. No. 1 — 

Cold Blast 

Milton, No. 1 — 

Hot Blast 

Muirkirk, No. 1— 

Cold Blast 

Hot Blast 

Morris Stirling's 2d quality. 
Gun Metal— 



455.93 

451.81 



442.43 
437.37 



434.06 
435.50 



439.37 
436.00 



444.56 
434.56 



29,107 H. 

17,466 H. 
13,434 H. 

18,855 H. 
16,676 H. 



145,435 H. 



,366 H. 
,397 H. 



81,770 H. 
82,739 H. 



American , 

Extra Specimens . 

Steel. 



595.00 



from 



Hammered Cast Steel, 

F. Krupp 

Tempered 

Bessemer Steel, from Hogbo. 

marked 10 

Bessemer Steel, Eng. Mean of 

four Experiments 

Naylor, Vickers & Co. Cruci 

ble Steel 

Mushet's Steel — 

Soft 

Cast Steel- 
Soft 

Not Hardened 

Mean Temper 

Razor Tempered 

Steel Wire Rope — 

Fine Wire 

Chrome Steel , 



485.37 
4S8.70 
492.50 
486.25 



490.00 



Wrought Iron. 



English 

In Bars 

Hammered 

Russian 

Swedish, in bars 

English, in wire 1-10 

diam . 

Russian, in wiro. : diam. 1-20 

to 1-30 inch . ... , 



inch 



481.20 
475.50 
487.00 



25,764 



14,000 
to 
34,000 Wa 
45,970 Wa. 



91,000 > c 
122,000 J k 
171,000 S. 

140,945 S. 

88,415 F. 
108,099 F. 

93,616 F. 
120,000 



150,000 



40.000 Ro. 
195,000 



57,300 La. 
57,300 La. 

67,200 Bru, 

60,480 La. 

71,680 R. 

80,000 To. 

96,000 Te. 
134,000 La. 
203,000 La. 



119,000 



36,288 H. s 
43,497 H. ! 

37,503 H. : 
35,316 H. 1 

33,453 F.i 
33,696 H. : 

34,587 F.i 

29,889 F.< 

36,693 F.' 
33,850 F.> 



225,568 
225,568 



198,944 
391,985 



Wa. 
Wa. 



22,907,700 H. 
22,473,650 H. 

15,381,200 H. 
13,730,500 H. 

14,313,500 F. 
14,322,500 F. 

13,981,000 F. 

11,974,500 F. 

14,003,550 F. 
13,294,490 F. 



27,548,000 Wa 



31,359,000 S. 



31, 



29,000,000 



819,000 S: 
215,000 F. 
278,000 F. 
901,000 F. 



oo 



312 



APPENDIX. 
TABLE.— Continued. 



Names op Materials. 



Wrought Iron. 

Rolled in sheets and cut cross- 
wise 

Cut lengthwise 

In chains, oval links, iron % 

in. diam 

Wire, American 

Lake Superior and Iron 

Mountain Charcoal Bloom. 

Missouri Iron, bar 

Tennessee, bar, 21 exp 

Salisbury, Ct., 40 exp 

Centre Co., Pa., 15 exp 

Phillipsburgh Wire, Pa. 

I 0.333 in., 13 exp. . . 

Diam.-< 0.190 in., 5 exp.. . . 
I 0.156 in., 5 exp.... 

Mean of 188 rolled bars 

Mean of 167 plates length- 



Mean of 160 plates crosswise 
Low Moor, bars , 









En .3 



Swedish, forged 

Hammered Bessemer Iron, 
from Hogbo 

Low Moor Rolled Puddled 
Iron 

Rolled Iron, Swedish, char- 
coal heath 



Lead, cast, English. 
Lead Wire 



Silver, standard . 

Tin, cast 

Zinc 



Stone— Natural and Art* 
ficial. 



Aberdeen, blue 

Cornish 

Killincy, very felspathic. 
Mount Sorrell, granite. . . 



Sandstones. 



Caithness Pavement 

Dundee Sandstone 

Derby Grit, a red, friable 

Sandstone 

Do. from another quarry. . 

Limestones. 

Limestone, Magnesian (Graf- 
ton, 111.) 



717.45 
705.12 

644.50 

455.68 

439.25 



164 
166 



166 



158 



148 
156 



\ 



40,320 Mi. 
31,360 Mi. 

48,160 Br. 
73,600 Ha. 

90,000 Ha. 
47,909 J. 
52,099 J. 
58,009 J. 
58,400 J. 

84,186 J. 
73,888 J. 
89,162 J. 
57,557 Ki. 

50,737 Ki. 

46,171 Ki. 

60,364 Ki. 
| 41,000 Ki. 
) 50,000 Ki. 



65,000 S. 

1,824 Re. 
2,581 M. 

40,902 M. 

5,322 M. 



§3 

a 1 ! ■ 

IIS 



o 

m oJPh 

It 



11 



32,320,000 S. 
31,976,000 S. 
27,000,000 S. 



4,008,000 Tr. 
13,680,000 Tr. 



10,914 Re. 
6,356 Re. 
10,780 Wn. 
12,286 F. 



6,493 Bn. 
6,630 Re. 

3,142 Re. 
4,345 Re. 



17 000 E. 



Same as Wt 
Iron. E. 



APPENDIX. 
TABLK.— Continued. 



311 



Names of Materials. 


■S.S 

■** ts 
■a - 

'&% 


S3 1 
m 


Cmshing Force 
per square inch 
in lbs. 

C. 




3 


Limestones. 


162 




7,713 Re. 

18,043 Wn. 
501 Re. 

3,216 Re. 

9,681 Ee. 
9,219 Ee. 
3,792 Ee. 
10,943 Wn. 

17,220 Wn. 

25,500 Ma. 

14,000 Ma. 

2,010 Re. 

808 Re. 

562 Ee. 

j 800 to 

"j 4,000 Ha. 

1,717 Ee. 

2,177 Ha. 

521 CI. 
j 500 to 
( 800 Ha. 

\ 

j 1,000 to 
15,900 G. 

334 Ee. 


1,062 
2,664 

\ 

\ 

\ 

11,202 B. 

j- 12,156 B. 

20,886 B. 
9,336 B. 




Limestone, Kerry, Listowel 






Chalk 








Other Stones. 

Alabaster (Oriental), white . . 
Marble, statuary 


170 








165 
168 
151 




25,200,000 T. 


Do. black G-alloway 

Portland Stone (Oolite) . 

Valentia, Kerry (slate stone). 
Green Stone, from Giant's 

Causeway 

Quartz Eock, Holyhead 


















Quartz Eock (parallel to lami- 










120 
158 

135.5 
130.31 


280 
300 








Artiflcial Stone. 
Brick, red 




















Brick, Stock 








Bricks set in cement (bricks 
















Cement, Portland, with sand. 
Cement, Portland, with no 




j 92 to 
{ 284 D. 

J711 










Chalk 


116.81 
153.31 

107 

47.37 
50.00 
49.56 
43.12 
55.81 
51 37 
67.51 
53.37 
45.12 






Glass, Plate 


9,420 
50 

16,000 Be. 
14,186 M. 
19,500 Be. 

j- 17,207 B. 

12,396 






j 120 to 
\ 240 Ha. 




Timber. 
Acacia, English 


1,152,000 B. 


Alder 


6,859 H. 

j 8,683 H. 

| 9.363 H. 

7,158 H. 






. a y. ( Ordinary state 

A8n 1 Very dry 

Bay Tree 


1,644,800 B. 


Bean, Tonquin 


2,601,600 B. 




15,784 B. 
17,850 B. 


7,733 H. ) 
9,363 H. f 


366011 i very Dry::::::::::: 


1,353,600 B. 



314 



APPENDIX. 
TABLE.— Continued. 



Names of Materials. 



$• a 

10 g 



£S 



g* 






o S..S 



g2« 

"3 -3 
■3 p. 



8#j4 
*s.g 

1-S 



Timber. 



Birch, common 

Birch, American 

Box, dry 

Bullet Tree (Berbice).. 

Cane . . . , 

Cedar, Canadian 

Crab Tree 

Deal- 
Christiana Middle. 
Norway Spruce... 

English 

Red 

White 

Elder 

Elm, seasoned 

Fir- 
New England 

Riga 

Hazel 

Lance "Wood 

Larch — 

G-reen 

Dry 

Lignum- vitee 

Mahogany, Spanish . . . 

Maple, Norway 

Oak- 



English. . . 
Canadian 



Dantzic. 

Adriatic , 

African Middle . . . 

Pear Tree 

Pine- 
Pitch 

Red 

American Yellow. 

Plum Tree 



Poplar 

Teak, dry.. 

Walnut 

Willow, dry. 



15,000 



40.50 
60.00 
64.31 
25.00 
56.81 
47.80 

43.62 
21.25 
29.37 



19,891 B. 



6,300 Be. 
11,400 Be. 



12,400 
17,600 
7,000 



43.43 
36.75 

34.56 
47.06 

53.75 

63.87 



35.00 
76.25 
50.00 
49.56 

58.37 

54.50 

47.24 
62.06 
60.75 
41.31 



10,230 
13.489 M. 



11,549 to 
12,857 B. 
18,000 Be. 
24,696 

10,220 B. 
8.900 B. 
11,800 M. 
16,500 
10.584 

17,300 M. 

10,253 

12,780 



41.25 
41.06 

28.81 

49.06 



41.06 
41.93 
34.37 



7,818 M. 



11,351 

7,200 

15.000 B. 
8,130 M. 
14,000 Be. 



j 4,533 H. 
1 6,402 H. 
11,663 H. 
10,299 H. 



5,674 H. 
6,499 H. 



5,748 H. 
6,741 H. 
8,467 H. 
10,331 H. 



5,748 to 
6,586 H. 



3,201 H. 
5,568 H. 

8,198 H. 



4,684 to 
9,509 H. 
4,231 to 
9,509 H. 



7,518 H. 



5,375 H. 
5,445 H. 
3,657 to 
9,367 H. 
3,107 to 
5,124 H. 
12,101 H. 
6,635 H. 



10,920 B. 
9,624 B. 

15,636 B. 



9,864 B. 



6,078 B. 

6,612 B. 
6,648 B. 
7,572 B. 



4,992 B. 
6,894 B. 



10,032 B. 

10,596 B. 

8,742 B. 

8,298 B. 

13,566 B. 



9,792 
8,946 B. 



14,722 B. 



1,562,400 B. 
1,257,600 B. 

2,610,600 B. 



1,672,000 B. 



699,840 B. 

2,191,200 B. 
1,328,800 
869,600 B. 



897,600 B. 
1,052,800 B. 



1,451,200 B. 

2,148,800 B. 

1,191,200 B. 

974,400 B. 

2,283,200 B. 



1,225,600 B. 
1,840,000 B. 
1,60C,000 Tr. 



2,414.400 B. 
306,000 



AUG 10 1W3 



